LMFDB - Embedded newform 75.10.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,10,Mod(1,75)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	75.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$38.6276877123$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 1546x^{2} + 152x + 559560 $ x^4 - 2x^3 - 1546x^2 + 152*x + 559560
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3\cdot 5^{2}\cdot 23 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-27.5964$ of defining polynomial
Character	$\chi$	$=$	75.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-38.7320 q^{2} -81.0000 q^{3} +988.165 q^{4} +3137.29 q^{6} -6373.44 q^{7} -18442.8 q^{8} +6561.00 q^{9} +O(q^{10})$	\(q-38.7320 q^{2} -81.0000 q^{3} +988.165 q^{4} +3137.29 q^{6} -6373.44 q^{7} -18442.8 q^{8} +6561.00 q^{9} +67637.2 q^{11} -80041.4 q^{12} +162029. q^{13} +246856. q^{14} +208386. q^{16} -308578. q^{17} -254120. q^{18} -526216. q^{19} +516249. q^{21} -2.61972e6 q^{22} -1.35822e6 q^{23} +1.49387e6 q^{24} -6.27569e6 q^{26} -531441. q^{27} -6.29801e6 q^{28} +6.02016e6 q^{29} -7.89958e6 q^{31} +1.37153e6 q^{32} -5.47861e6 q^{33} +1.19518e7 q^{34} +6.48335e6 q^{36} -6.52067e6 q^{37} +2.03814e7 q^{38} -1.31243e7 q^{39} -3.65494e6 q^{41} -1.99953e7 q^{42} -2.63401e7 q^{43} +6.68367e7 q^{44} +5.26067e7 q^{46} +1.04707e7 q^{47} -1.68792e7 q^{48} +267157. q^{49} +2.49948e7 q^{51} +1.60111e8 q^{52} +1.10225e8 q^{53} +2.05838e7 q^{54} +1.17544e8 q^{56} +4.26235e7 q^{57} -2.33173e8 q^{58} +2.16362e6 q^{59} -1.44173e8 q^{61} +3.05966e8 q^{62} -4.18162e7 q^{63} -1.59816e8 q^{64} +2.12197e8 q^{66} +1.09836e8 q^{67} -3.04926e8 q^{68} +1.10016e8 q^{69} +1.72652e8 q^{71} -1.21003e8 q^{72} +2.71753e8 q^{73} +2.52559e8 q^{74} -5.19988e8 q^{76} -4.31082e8 q^{77} +5.08331e8 q^{78} +6.32088e8 q^{79} +4.30467e7 q^{81} +1.41563e8 q^{82} -3.09787e7 q^{83} +5.10139e8 q^{84} +1.02020e9 q^{86} -4.87633e8 q^{87} -1.24742e9 q^{88} -3.96023e7 q^{89} -1.03268e9 q^{91} -1.34215e9 q^{92} +6.39866e8 q^{93} -4.05552e8 q^{94} -1.11094e8 q^{96} +9.30586e8 q^{97} -1.03475e7 q^{98} +4.43767e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 324 q^{3} + 1792 q^{4} - 162 q^{6} - 13036 q^{7} + 24636 q^{8} + 26244 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 - 324 * q^3 + 1792 * q^4 - 162 * q^6 - 13036 * q^7 + 24636 * q^8 + 26244 * q^9	\( 4 q + 2 q^{2} - 324 q^{3} + 1792 q^{4} - 162 q^{6} - 13036 q^{7} + 24636 q^{8} + 26244 q^{9} + 104696 q^{11} - 145152 q^{12} - 140812 q^{13} - 181062 q^{14} + 1319800 q^{16} - 489352 q^{17} + 13122 q^{18} + 393308 q^{19} + 1055916 q^{21} + 793956 q^{22} - 2326488 q^{23} - 1995516 q^{24} - 6477502 q^{26} - 2125764 q^{27} - 10094272 q^{28} + 4926616 q^{29} - 97516 q^{31} + 29228344 q^{32} - 8480376 q^{33} + 16828644 q^{34} + 11757312 q^{36} + 4958984 q^{37} + 43844342 q^{38} + 11405772 q^{39} - 996656 q^{41} + 14666022 q^{42} - 28298860 q^{43} + 206545216 q^{44} + 103917804 q^{46} + 30714920 q^{47} - 106903800 q^{48} + 80698920 q^{49} + 39637512 q^{51} + 146870528 q^{52} - 70694368 q^{53} - 1062882 q^{54} + 170654220 q^{56} - 31857948 q^{57} - 532031436 q^{58} + 225946712 q^{59} + 6295340 q^{61} + 665342778 q^{62} - 85529196 q^{63} + 632883328 q^{64} - 64310436 q^{66} + 217434788 q^{67} - 603488576 q^{68} + 188445528 q^{69} + 22716688 q^{71} + 161636796 q^{72} - 79864888 q^{73} + 193190788 q^{74} + 641703872 q^{76} - 395948232 q^{77} + 524677662 q^{78} + 1276962080 q^{79} + 172186884 q^{81} + 186865488 q^{82} - 175482984 q^{83} + 817636032 q^{84} + 1426174466 q^{86} - 399055896 q^{87} + 2581163256 q^{88} - 897754752 q^{89} - 1837903676 q^{91} + 1998210624 q^{92} + 7898796 q^{93} - 2190567228 q^{94} - 2367495864 q^{96} - 1702783612 q^{97} - 591217148 q^{98} + 686910456 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 324 * q^3 + 1792 * q^4 - 162 * q^6 - 13036 * q^7 + 24636 * q^8 + 26244 * q^9 + 104696 * q^11 - 145152 * q^12 - 140812 * q^13 - 181062 * q^14 + 1319800 * q^16 - 489352 * q^17 + 13122 * q^18 + 393308 * q^19 + 1055916 * q^21 + 793956 * q^22 - 2326488 * q^23 - 1995516 * q^24 - 6477502 * q^26 - 2125764 * q^27 - 10094272 * q^28 + 4926616 * q^29 - 97516 * q^31 + 29228344 * q^32 - 8480376 * q^33 + 16828644 * q^34 + 11757312 * q^36 + 4958984 * q^37 + 43844342 * q^38 + 11405772 * q^39 - 996656 * q^41 + 14666022 * q^42 - 28298860 * q^43 + 206545216 * q^44 + 103917804 * q^46 + 30714920 * q^47 - 106903800 * q^48 + 80698920 * q^49 + 39637512 * q^51 + 146870528 * q^52 - 70694368 * q^53 - 1062882 * q^54 + 170654220 * q^56 - 31857948 * q^57 - 532031436 * q^58 + 225946712 * q^59 + 6295340 * q^61 + 665342778 * q^62 - 85529196 * q^63 + 632883328 * q^64 - 64310436 * q^66 + 217434788 * q^67 - 603488576 * q^68 + 188445528 * q^69 + 22716688 * q^71 + 161636796 * q^72 - 79864888 * q^73 + 193190788 * q^74 + 641703872 * q^76 - 395948232 * q^77 + 524677662 * q^78 + 1276962080 * q^79 + 172186884 * q^81 + 186865488 * q^82 - 175482984 * q^83 + 817636032 * q^84 + 1426174466 * q^86 - 399055896 * q^87 + 2581163256 * q^88 - 897754752 * q^89 - 1837903676 * q^91 + 1998210624 * q^92 + 7898796 * q^93 - 2190567228 * q^94 - 2367495864 * q^96 - 1702783612 * q^97 - 591217148 * q^98 + 686910456 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−38.7320	−1.71173	−0.855864	−	0.517202i	$-0.826974\pi$
$2$	−38.7320	−1.71173	−0.855864	+	0.517202i	$0.826974\pi$
$3$	−81.0000	−0.577350
$3$	−81.0000	−0.577350
$4$	988.165	1.93001
$5$	0	0
$5$	0	0
$6$	3137.29	0.988266
$7$	−6373.44	−1.00330	−0.501652	−	0.865069i	$-0.667274\pi$
$7$	−6373.44	−1.00330	−0.501652	+	0.865069i	$0.667274\pi$
$8$	−18442.8	−1.59192
$9$	6561.00	0.333333
$10$	0	0
$11$	67637.2	1.39289	0.696447	−	0.717608i	$-0.254763\pi$
$11$	67637.2	1.39289	0.696447	+	0.717608i	$0.254763\pi$
$12$	−80041.4	−1.11429
$13$	162029.	1.57343	0.786715	−	0.617317i	$-0.211780\pi$
$13$	162029.	1.57343	0.786715	+	0.617317i	$0.211780\pi$
$14$	246856.	1.71738
$15$	0	0
$16$	208386.	0.794928
$17$	−308578.	−0.896077	−0.448038	−	0.894014i	$-0.647877\pi$
$17$	−308578.	−0.896077	−0.448038	+	0.894014i	$0.647877\pi$
$18$	−254120.	−0.570576
$19$	−526216.	−0.926345	−0.463173	−	0.886268i	$-0.653289\pi$
$19$	−526216.	−0.926345	−0.463173	+	0.886268i	$0.653289\pi$
$20$	0	0
$21$	516249.	0.579258
$22$	−2.61972e6	−2.38426
$23$	−1.35822e6	−1.01204	−0.506018	−	0.862523i	$-0.668883\pi$
$23$	−1.35822e6	−1.01204	−0.506018	+	0.862523i	$0.668883\pi$
$24$	1.49387e6	0.919097
$25$	0	0
$26$	−6.27569e6	−2.69328
$27$	−531441.	−0.192450
$28$	−6.29801e6	−1.93639
$29$	6.02016e6	1.58058	0.790291	−	0.612732i	$-0.209929\pi$
$29$	6.02016e6	1.58058	0.790291	+	0.612732i	$0.209929\pi$
$30$	0	0
$31$	−7.89958e6	−1.53630	−0.768150	−	0.640270i	$-0.778823\pi$
$31$	−7.89958e6	−1.53630	−0.768150	+	0.640270i	$0.778823\pi$
$32$	1.37153e6	0.231223
$33$	−5.47861e6	−0.804188
$34$	1.19518e7	1.53384
$35$	0	0
$36$	6.48335e6	0.643337
$37$	−6.52067e6	−0.571985	−0.285993	−	0.958232i	$-0.592323\pi$
$37$	−6.52067e6	−0.571985	−0.285993	+	0.958232i	$0.592323\pi$
$38$	2.03814e7	1.58565
$39$	−1.31243e7	−0.908420
$40$	0	0
$41$	−3.65494e6	−0.202001	−0.101000	−	0.994886i	$-0.532204\pi$
$41$	−3.65494e6	−0.202001	−0.101000	+	0.994886i	$0.532204\pi$
$42$	−1.99953e7	−0.991532
$43$	−2.63401e7	−1.17492	−0.587461	−	0.809252i	$-0.699872\pi$
$43$	−2.63401e7	−1.17492	−0.587461	+	0.809252i	$0.699872\pi$
$44$	6.68367e7	2.68830
$45$	0	0
$46$	5.26067e7	1.73233
$47$	1.04707e7	0.312994	0.156497	−	0.987678i	$-0.449980\pi$
$47$	1.04707e7	0.312994	0.156497	+	0.987678i	$0.449980\pi$
$48$	−1.68792e7	−0.458952
$49$	267157.	0.00662039
$50$	0	0
$51$	2.49948e7	0.517350
$52$	1.60111e8	3.03673
$53$	1.10225e8	1.91884	0.959422	−	0.281973i	$-0.0909888\pi$
$53$	1.10225e8	1.91884	0.959422	+	0.281973i	$0.0909888\pi$
$54$	2.05838e7	0.329422
$55$	0	0
$56$	1.17544e8	1.59718
$57$	4.26235e7	0.534826
$58$	−2.33173e8	−2.70552
$59$	2.16362e6	0.0232460	0.0116230	−	0.999932i	$-0.496300\pi$
$59$	2.16362e6	0.0232460	0.0116230	+	0.999932i	$0.496300\pi$
$60$	0	0
$61$	−1.44173e8	−1.33321	−0.666607	−	0.745409i	$-0.732254\pi$
$61$	−1.44173e8	−1.33321	−0.666607	+	0.745409i	$0.732254\pi$
$62$	3.05966e8	2.62973
$63$	−4.18162e7	−0.334435
$64$	−1.59816e8	−1.19072
$65$	0	0
$66$	2.12197e8	1.37655
$67$	1.09836e8	0.665896	0.332948	−	0.942945i	$-0.391957\pi$
$67$	1.09836e8	0.665896	0.332948	+	0.942945i	$0.391957\pi$
$68$	−3.04926e8	−1.72944
$69$	1.10016e8	0.584299
$70$	0	0
$71$	1.72652e8	0.806323	0.403162	−	0.915129i	$-0.367911\pi$
$71$	1.72652e8	0.806323	0.403162	+	0.915129i	$0.367911\pi$
$72$	−1.21003e8	−0.530641
$73$	2.71753e8	1.12001	0.560005	−	0.828489i	$-0.310799\pi$
$73$	2.71753e8	1.12001	0.560005	+	0.828489i	$0.310799\pi$
$74$	2.52559e8	0.979083
$75$	0	0
$76$	−5.19988e8	−1.78786
$77$	−4.31082e8	−1.39750
$78$	5.08331e8	1.55497
$79$	6.32088e8	1.82581	0.912905	−	0.408171i	$-0.133833\pi$
$79$	6.32088e8	1.82581	0.912905	+	0.408171i	$0.133833\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	1.41563e8	0.345771
$83$	−3.09787e7	−0.0716493	−0.0358247	−	0.999358i	$-0.511406\pi$
$83$	−3.09787e7	−0.0716493	−0.0358247	+	0.999358i	$0.511406\pi$
$84$	5.10139e8	1.11797
$85$	0	0
$86$	1.02020e9	2.01115
$87$	−4.87633e8	−0.912549
$88$	−1.24742e9	−2.21738
$89$	−3.96023e7	−0.0669060	−0.0334530	−	0.999440i	$-0.510650\pi$
$89$	−3.96023e7	−0.0669060	−0.0334530	+	0.999440i	$0.510650\pi$
$90$	0	0
$91$	−1.03268e9	−1.57863
$92$	−1.34215e9	−1.95324
$93$	6.39866e8	0.886984
$94$	−4.05552e8	−0.535761
$95$	0	0
$96$	−1.11094e8	−0.133497
$97$	9.30586e8	1.06729	0.533646	−	0.845708i	$-0.320821\pi$
$97$	9.30586e8	1.06729	0.533646	+	0.845708i	$0.320821\pi$
$98$	−1.03475e7	−0.0113323
$99$	4.43767e8	0.464298
$100$	0	0
$101$	−9.63133e8	−0.920958	−0.460479	−	0.887671i	$-0.652322\pi$
$101$	−9.63133e8	−0.920958	−0.460479	+	0.887671i	$0.652322\pi$
$102$	−9.68099e8	−0.885562
$103$	−1.77100e8	−0.155043	−0.0775214	−	0.996991i	$-0.524701\pi$
$103$	−1.77100e8	−0.155043	−0.0775214	+	0.996991i	$0.524701\pi$
$104$	−2.98827e9	−2.50478
$105$	0	0
$106$	−4.26924e9	−3.28454
$107$	6.20403e7	0.0457558	0.0228779	−	0.999738i	$-0.492717\pi$
$107$	6.20403e7	0.0457558	0.0228779	+	0.999738i	$0.492717\pi$
$108$	−5.25151e8	−0.371431
$109$	−9.82841e8	−0.666905	−0.333453	−	0.942767i	$-0.608214\pi$
$109$	−9.82841e8	−0.666905	−0.333453	+	0.942767i	$0.608214\pi$
$110$	0	0
$111$	5.28175e8	0.330236
$112$	−1.32813e9	−0.797555
$113$	2.49961e9	1.44218	0.721088	−	0.692843i	$-0.243642\pi$
$113$	2.49961e9	1.44218	0.721088	+	0.692843i	$0.243642\pi$
$114$	−1.65089e9	−0.915476
$115$	0	0
$116$	5.94891e9	3.05054
$117$	1.06307e9	0.524476
$118$	−8.38013e7	−0.0397907
$119$	1.96671e9	0.899038
$120$	0	0
$121$	2.21684e9	0.940156
$122$	5.58411e9	2.28210
$123$	2.96051e8	0.116625
$124$	−7.80608e9	−2.96507
$125$	0	0
$126$	1.61962e9	0.572461
$127$	−1.37360e8	−0.0468538	−0.0234269	−	0.999726i	$-0.507458\pi$
$127$	−1.37360e8	−0.0468538	−0.0234269	+	0.999726i	$0.507458\pi$
$128$	5.48775e9	1.80696
$129$	2.13355e9	0.678342
$130$	0	0
$131$	−3.52545e9	−1.04591	−0.522955	−	0.852360i	$-0.675170\pi$
$131$	−3.52545e9	−1.04591	−0.522955	+	0.852360i	$0.675170\pi$
$132$	−5.41377e9	−1.55209
$133$	3.35381e9	0.929407
$134$	−4.25414e9	−1.13983
$135$	0	0
$136$	5.69105e9	1.42649
$137$	3.50018e9	0.848882	0.424441	−	0.905456i	$-0.360471\pi$
$137$	3.50018e9	0.848882	0.424441	+	0.905456i	$0.360471\pi$
$138$	−4.26114e9	−1.00016
$139$	4.30084e9	0.977208	0.488604	−	0.872506i	$-0.337506\pi$
$139$	4.30084e9	0.977208	0.488604	+	0.872506i	$0.337506\pi$
$140$	0	0
$141$	−8.48129e8	−0.180707
$142$	−6.68715e9	−1.38021
$143$	1.09592e10	2.19162
$144$	1.36722e9	0.264976
$145$	0	0
$146$	−1.05255e10	−1.91715
$147$	−2.16397e7	−0.00382229
$148$	−6.44350e9	−1.10394
$149$	−2.49219e9	−0.414232	−0.207116	−	0.978316i	$-0.566408\pi$
$149$	−2.49219e9	−0.414232	−0.207116	+	0.978316i	$0.566408\pi$
$150$	0	0
$151$	−2.62969e9	−0.411631	−0.205816	−	0.978591i	$-0.565985\pi$
$151$	−2.62969e9	−0.411631	−0.205816	+	0.978591i	$0.565985\pi$
$152$	9.70490e9	1.47467
$153$	−2.02458e9	−0.298692
$154$	1.66966e10	2.39214
$155$	0	0
$156$	−1.29690e10	−1.75326
$157$	−8.41289e8	−0.110509	−0.0552544	−	0.998472i	$-0.517597\pi$
$157$	−8.41289e8	−0.110509	−0.0552544	+	0.998472i	$0.517597\pi$
$158$	−2.44820e10	−3.12529
$159$	−8.92825e9	−1.10785
$160$	0	0
$161$	8.65656e9	1.01538
$162$	−1.66728e9	−0.190192
$163$	2.91546e8	0.0323491	0.0161746	−	0.999869i	$-0.494851\pi$
$163$	2.91546e8	0.0323491	0.0161746	+	0.999869i	$0.494851\pi$
$164$	−3.61169e9	−0.389864
$165$	0	0
$166$	1.19987e9	0.122644
$167$	−3.59977e9	−0.358138	−0.179069	−	0.983836i	$-0.557309\pi$
$167$	−3.59977e9	−0.358138	−0.179069	+	0.983836i	$0.557309\pi$
$168$	−9.52108e9	−0.922134
$169$	1.56488e10	1.47568
$170$	0	0
$171$	−3.45250e9	−0.308782
$172$	−2.60283e10	−2.26761
$173$	−8.47314e9	−0.719179	−0.359589	−	0.933111i	$-0.617083\pi$
$173$	−8.47314e9	−0.719179	−0.359589	+	0.933111i	$0.617083\pi$
$174$	1.88870e10	1.56204
$175$	0	0
$176$	1.40946e10	1.10725
$177$	−1.75253e8	−0.0134211
$178$	1.53387e9	0.114525
$179$	1.15470e10	0.840683	0.420342	−	0.907366i	$-0.361910\pi$
$179$	1.15470e10	0.840683	0.420342	+	0.907366i	$0.361910\pi$
$180$	0	0
$181$	2.19482e10	1.52001	0.760003	−	0.649920i	$-0.225197\pi$
$181$	2.19482e10	1.52001	0.760003	+	0.649920i	$0.225197\pi$
$182$	3.99978e10	2.70218
$183$	1.16780e10	0.769732
$184$	2.50495e10	1.61108
$185$	0	0
$186$	−2.47833e10	−1.51827
$187$	−2.08714e10	−1.24814
$188$	1.03468e10	0.604082
$189$	3.38711e9	0.193086
$190$	0	0
$191$	3.31268e10	1.80107	0.900533	−	0.434787i	$-0.143176\pi$
$191$	3.31268e10	1.80107	0.900533	+	0.434787i	$0.143176\pi$
$192$	1.29451e10	0.687462
$193$	2.44518e10	1.26854	0.634269	−	0.773113i	$-0.281301\pi$
$193$	2.44518e10	1.26854	0.634269	+	0.773113i	$0.281301\pi$
$194$	−3.60434e10	−1.82691
$195$	0	0
$196$	2.63995e8	0.0127774
$197$	−1.84287e10	−0.871759	−0.435880	−	0.900005i	$-0.643563\pi$
$197$	−1.84287e10	−0.871759	−0.435880	+	0.900005i	$0.643563\pi$
$198$	−1.71880e10	−0.794752
$199$	−2.17469e10	−0.983013	−0.491506	−	0.870874i	$-0.663554\pi$
$199$	−2.17469e10	−0.983013	−0.491506	+	0.870874i	$0.663554\pi$
$200$	0	0
$201$	−8.89668e9	−0.384455
$202$	3.73040e10	1.57643
$203$	−3.83691e10	−1.58581
$204$	2.46990e10	0.998491
$205$	0	0
$206$	6.85944e9	0.265391
$207$	−8.91131e9	−0.337345
$208$	3.37645e10	1.25076
$209$	−3.55918e10	−1.29030
$210$	0	0
$211$	3.17325e10	1.10213	0.551066	−	0.834462i	$-0.314221\pi$
$211$	3.17325e10	1.10213	0.551066	+	0.834462i	$0.314221\pi$
$212$	1.08921e11	3.70339
$213$	−1.39848e10	−0.465531
$214$	−2.40294e9	−0.0783215
$215$	0	0
$216$	9.80126e9	0.306366
$217$	5.03475e10	1.54138
$218$	3.80674e10	1.14156
$219$	−2.20120e10	−0.646638
$220$	0	0
$221$	−4.99986e10	−1.40991
$222$	−2.04572e10	−0.565274
$223$	−3.98798e10	−1.07989	−0.539947	−	0.841699i	$-0.681556\pi$
$223$	−3.98798e10	−1.07989	−0.539947	+	0.841699i	$0.681556\pi$
$224$	−8.74140e9	−0.231988
$225$	0	0
$226$	−9.68146e10	−2.46861
$227$	8.18137e9	0.204508	0.102254	−	0.994758i	$-0.467395\pi$
$227$	8.18137e9	0.204508	0.102254	+	0.994758i	$0.467395\pi$
$228$	4.21191e10	1.03222
$229$	4.29758e10	1.03268	0.516339	−	0.856385i	$-0.327295\pi$
$229$	4.29758e10	1.03268	0.516339	+	0.856385i	$0.327295\pi$
$230$	0	0
$231$	3.49176e10	0.806846
$232$	−1.11029e11	−2.51616
$233$	5.73139e10	1.27397	0.636983	−	0.770878i	$-0.280182\pi$
$233$	5.73139e10	1.27397	0.636983	+	0.770878i	$0.280182\pi$
$234$	−4.11748e10	−0.897761
$235$	0	0
$236$	2.13802e9	0.0448649
$237$	−5.11991e10	−1.05413
$238$	−7.61744e10	−1.53891
$239$	1.90346e10	0.377359	0.188679	−	0.982039i	$-0.439579\pi$
$239$	1.90346e10	0.377359	0.188679	+	0.982039i	$0.439579\pi$
$240$	0	0
$241$	4.53326e10	0.865632	0.432816	−	0.901482i	$-0.357520\pi$
$241$	4.53326e10	0.865632	0.432816	+	0.901482i	$0.357520\pi$
$242$	−8.58625e10	−1.60929
$243$	−3.48678e9	−0.0641500
$244$	−1.42467e11	−2.57312
$245$	0	0
$246$	−1.14666e10	−0.199631
$247$	−8.52622e10	−1.45754
$248$	1.45690e11	2.44567
$249$	2.50928e9	0.0413668
$250$	0	0
$251$	9.96236e10	1.58427	0.792137	−	0.610343i	$-0.208968\pi$
$251$	9.96236e10	1.58427	0.792137	+	0.610343i	$0.208968\pi$
$252$	−4.13213e10	−0.645463
$253$	−9.18664e10	−1.40966
$254$	5.32024e9	0.0802009
$255$	0	0
$256$	−1.30726e11	−1.90231
$257$	−1.26491e11	−1.80867	−0.904337	−	0.426819i	$-0.859634\pi$
$257$	−1.26491e11	−1.80867	−0.904337	+	0.426819i	$0.859634\pi$
$258$	−8.26364e10	−1.16114
$259$	4.15591e10	0.573875
$260$	0	0
$261$	3.94983e10	0.526861
$262$	1.36548e11	1.79031
$263$	4.01277e10	0.517182	0.258591	−	0.965987i	$-0.416742\pi$
$263$	4.01277e10	0.517182	0.258591	+	0.965987i	$0.416742\pi$
$264$	1.01041e11	1.28021
$265$	0	0
$266$	−1.29900e11	−1.59089
$267$	3.20778e9	0.0386282
$268$	1.08536e11	1.28518
$269$	5.17457e10	0.602544	0.301272	−	0.953538i	$-0.402589\pi$
$269$	5.17457e10	0.602544	0.301272	+	0.953538i	$0.402589\pi$
$270$	0	0
$271$	−1.59741e10	−0.179909	−0.0899546	−	0.995946i	$-0.528672\pi$
$271$	−1.59741e10	−0.179909	−0.0899546	+	0.995946i	$0.528672\pi$
$272$	−6.43033e10	−0.712316
$273$	8.36472e10	0.911422
$274$	−1.35569e11	−1.45305
$275$	0	0
$276$	1.08714e11	1.12770
$277$	−5.11479e10	−0.521998	−0.260999	−	0.965339i	$-0.584052\pi$
$277$	−5.11479e10	−0.521998	−0.260999	+	0.965339i	$0.584052\pi$
$278$	−1.66580e11	−1.67271
$279$	−5.18291e10	−0.512100
$280$	0	0
$281$	1.33633e10	0.127860	0.0639302	−	0.997954i	$-0.479636\pi$
$281$	1.33633e10	0.127860	0.0639302	+	0.997954i	$0.479636\pi$
$282$	3.28497e10	0.309322
$283$	−1.02607e11	−0.950909	−0.475454	−	0.879740i	$-0.657716\pi$
$283$	−1.02607e11	−0.950909	−0.475454	+	0.879740i	$0.657716\pi$
$284$	1.70609e11	1.55621
$285$	0	0
$286$	−4.24470e11	−3.75146
$287$	2.32946e10	0.202669
$288$	8.99864e9	0.0770745
$289$	−2.33673e10	−0.197046
$290$	0	0
$291$	−7.53774e10	−0.616202
$292$	2.68537e11	2.16163
$293$	1.05032e11	0.832567	0.416283	−	0.909235i	$-0.363332\pi$
$293$	1.05032e11	0.832567	0.416283	+	0.909235i	$0.363332\pi$
$294$	8.38148e8	0.00654271
$295$	0	0
$296$	1.20260e11	0.910556
$297$	−3.59452e10	−0.268063
$298$	9.65275e10	0.709052
$299$	−2.20071e11	−1.59237
$300$	0	0
$301$	1.67877e11	1.17880
$302$	1.01853e11	0.704601
$303$	7.80138e10	0.531716
$304$	−1.09656e11	−0.736378
$305$	0	0
$306$	7.84160e10	0.511280
$307$	2.97435e11	1.91104	0.955520	−	0.294926i	$-0.0952948\pi$
$307$	2.97435e11	1.91104	0.955520	+	0.294926i	$0.0952948\pi$
$308$	−4.25980e11	−2.69718
$309$	1.43451e10	0.0895140
$310$	0	0
$311$	2.27383e11	1.37827	0.689137	−	0.724631i	$-0.257990\pi$
$311$	2.27383e11	1.37827	0.689137	+	0.724631i	$0.257990\pi$
$312$	2.42050e11	1.44613
$313$	−1.41984e10	−0.0836159	−0.0418079	−	0.999126i	$-0.513312\pi$
$313$	−1.41984e10	−0.0836159	−0.0418079	+	0.999126i	$0.513312\pi$
$314$	3.25848e10	0.189161
$315$	0	0
$316$	6.24607e11	3.52383
$317$	−7.94894e10	−0.442122	−0.221061	−	0.975260i	$-0.570952\pi$
$317$	−7.94894e10	−0.442122	−0.221061	+	0.975260i	$0.570952\pi$
$318$	3.45809e11	1.89633
$319$	4.07187e11	2.20158
$320$	0	0
$321$	−5.02526e9	−0.0264171
$322$	−3.35286e11	−1.73805
$323$	1.62379e11	0.830077
$324$	4.25373e10	0.214446
$325$	0	0
$326$	−1.12921e10	−0.0553729
$327$	7.96101e10	0.385038
$328$	6.74074e10	0.321570
$329$	−6.67346e10	−0.314029
$330$	0	0
$331$	−8.00352e9	−0.0366484	−0.0183242	−	0.999832i	$-0.505833\pi$
$331$	−8.00352e9	−0.0366484	−0.0183242	+	0.999832i	$0.505833\pi$
$332$	−3.06121e10	−0.138284
$333$	−4.27821e10	−0.190662
$334$	1.39426e11	0.613035
$335$	0	0
$336$	1.07579e11	0.460468
$337$	4.70459e11	1.98695	0.993476	−	0.114045i	$-0.0363808\pi$
$337$	4.70459e11	1.98695	0.993476	+	0.114045i	$0.0363808\pi$
$338$	−6.06110e11	−2.52596
$339$	−2.02468e11	−0.832641
$340$	0	0
$341$	−5.34305e11	−2.13991
$342$	1.33722e11	0.528550
$343$	2.55489e11	0.996662
$344$	4.85785e11	1.87039
$345$	0	0
$346$	3.28181e11	1.23104
$347$	−3.07958e11	−1.14027	−0.570137	−	0.821550i	$-0.693110\pi$
$347$	−3.07958e11	−1.14027	−0.570137	+	0.821550i	$0.693110\pi$
$348$	−4.81862e11	−1.76123
$349$	5.25038e11	1.89442	0.947210	−	0.320614i	$-0.103889\pi$
$349$	5.25038e11	1.89442	0.947210	+	0.320614i	$0.103889\pi$
$350$	0	0
$351$	−8.61088e10	−0.302807
$352$	9.27667e10	0.322070
$353$	1.20347e11	0.412525	0.206263	−	0.978497i	$-0.433870\pi$
$353$	1.20347e11	0.412525	0.206263	+	0.978497i	$0.433870\pi$
$354$	6.78791e9	0.0229732
$355$	0	0
$356$	−3.91336e10	−0.129129
$357$	−1.59303e11	−0.519060
$358$	−4.47240e11	−1.43902
$359$	1.79853e10	0.0571469	0.0285735	−	0.999592i	$-0.490904\pi$
$359$	1.79853e10	0.0571469	0.0285735	+	0.999592i	$0.490904\pi$
$360$	0	0
$361$	−4.57843e10	−0.141884
$362$	−8.50097e11	−2.60184
$363$	−1.79564e11	−0.542799
$364$	−1.02046e12	−3.04677
$365$	0	0
$366$	−4.52313e11	−1.31757
$367$	−4.57539e11	−1.31653	−0.658266	−	0.752786i	$-0.728710\pi$
$367$	−4.57539e11	−1.31653	−0.658266	+	0.752786i	$0.728710\pi$
$368$	−2.83034e11	−0.804496
$369$	−2.39801e10	−0.0673337
$370$	0	0
$371$	−7.02515e11	−1.92519
$372$	6.32293e11	1.71189
$373$	4.29004e11	1.14755	0.573776	−	0.819013i	$-0.305478\pi$
$373$	4.29004e11	1.14755	0.573776	+	0.819013i	$0.305478\pi$
$374$	8.08389e11	2.13648
$375$	0	0
$376$	−1.93110e11	−0.498263
$377$	9.75439e11	2.48693
$378$	−1.31189e11	−0.330511
$379$	−2.97456e11	−0.740537	−0.370268	−	0.928925i	$-0.620734\pi$
$379$	−2.97456e11	−0.740537	−0.370268	+	0.928925i	$0.620734\pi$
$380$	0	0
$381$	1.11262e10	0.0270511
$382$	−1.28307e12	−3.08293
$383$	3.76724e11	0.894600	0.447300	−	0.894384i	$-0.352386\pi$
$383$	3.76724e11	0.894600	0.447300	+	0.894384i	$0.352386\pi$
$384$	−4.44508e11	−1.04325
$385$	0	0
$386$	−9.47067e11	−2.17139
$387$	−1.72817e11	−0.391641
$388$	9.19572e11	2.05989
$389$	4.60594e11	1.01987	0.509935	−	0.860213i	$-0.329669\pi$
$389$	4.60594e11	1.01987	0.509935	+	0.860213i	$0.329669\pi$
$390$	0	0
$391$	4.19118e11	0.906862
$392$	−4.92712e9	−0.0105392
$393$	2.85562e11	0.603856
$394$	7.13779e11	1.49221
$395$	0	0
$396$	4.38515e11	0.896100
$397$	1.53795e11	0.310731	0.155365	−	0.987857i	$-0.450344\pi$
$397$	1.53795e11	0.310731	0.155365	+	0.987857i	$0.450344\pi$
$398$	8.42302e11	1.68265
$399$	−2.71658e11	−0.536593
$400$	0	0
$401$	2.49483e11	0.481827	0.240914	−	0.970547i	$-0.422553\pi$
$401$	2.49483e11	0.481827	0.240914	+	0.970547i	$0.422553\pi$
$402$	3.44586e11	0.658082
$403$	−1.27996e12	−2.41726
$404$	−9.51734e11	−1.77746
$405$	0	0
$406$	1.48611e12	2.71447
$407$	−4.41040e11	−0.796715
$408$	−4.60975e11	−0.823582
$409$	3.89226e11	0.687776	0.343888	−	0.939011i	$-0.388256\pi$
$409$	3.89226e11	0.687776	0.343888	+	0.939011i	$0.388256\pi$
$410$	0	0
$411$	−2.83514e11	−0.490102
$412$	−1.75004e11	−0.299234
$413$	−1.37897e10	−0.0233228
$414$	3.45152e11	0.577443
$415$	0	0
$416$	2.22228e11	0.363814
$417$	−3.48368e11	−0.564191
$418$	1.37854e12	2.20864
$419$	−2.70677e11	−0.429031	−0.214516	−	0.976721i	$-0.568817\pi$
$419$	−2.70677e11	−0.429031	−0.214516	+	0.976721i	$0.568817\pi$
$420$	0	0
$421$	2.39519e10	0.0371596	0.0185798	−	0.999827i	$-0.494086\pi$
$421$	2.39519e10	0.0371596	0.0185798	+	0.999827i	$0.494086\pi$
$422$	−1.22906e12	−1.88655
$423$	6.86984e10	0.104331
$424$	−2.03286e12	−3.05465
$425$	0	0
$426$	5.41659e11	0.796862
$427$	9.18879e11	1.33762
$428$	6.13060e10	0.0883092
$429$	−8.87693e11	−1.26533
$430$	0	0
$431$	−4.02812e11	−0.562283	−0.281142	−	0.959666i	$-0.590713\pi$
$431$	−4.02812e11	−0.562283	−0.281142	+	0.959666i	$0.590713\pi$
$432$	−1.10745e11	−0.152984
$433$	−3.35143e11	−0.458178	−0.229089	−	0.973405i	$-0.573575\pi$
$433$	−3.35143e11	−0.458178	−0.229089	+	0.973405i	$0.573575\pi$
$434$	−1.95006e12	−2.63842
$435$	0	0
$436$	−9.71209e11	−1.28713
$437$	7.14719e11	0.937495
$438$	8.52568e11	1.10687
$439$	7.00427e11	0.900062	0.450031	−	0.893013i	$-0.351413\pi$
$439$	7.00427e11	0.900062	0.450031	+	0.893013i	$0.351413\pi$
$440$	0	0
$441$	1.75282e9	0.00220680
$442$	1.93654e12	2.41339
$443$	1.48088e11	0.182686	0.0913429	−	0.995820i	$-0.470884\pi$
$443$	1.48088e11	0.182686	0.0913429	+	0.995820i	$0.470884\pi$
$444$	5.21924e11	0.637358
$445$	0	0
$446$	1.54462e12	1.84848
$447$	2.01868e11	0.239157
$448$	1.01858e12	1.19465
$449$	−7.97636e11	−0.926181	−0.463091	−	0.886311i	$-0.653260\pi$
$449$	−7.97636e11	−0.926181	−0.463091	+	0.886311i	$0.653260\pi$
$450$	0	0
$451$	−2.47210e11	−0.281366
$452$	2.47002e12	2.78342
$453$	2.13005e11	0.237655
$454$	−3.16881e11	−0.350062
$455$	0	0
$456$	−7.86097e11	−0.851401
$457$	−1.02541e12	−1.09970	−0.549851	−	0.835263i	$-0.685316\pi$
$457$	−1.02541e12	−1.09970	−0.549851	+	0.835263i	$0.685316\pi$
$458$	−1.66454e12	−1.76766
$459$	1.63991e11	0.172450
$460$	0	0
$461$	−6.71750e11	−0.692713	−0.346357	−	0.938103i	$-0.612581\pi$
$461$	−6.71750e11	−0.692713	−0.346357	+	0.938103i	$0.612581\pi$
$462$	−1.35243e12	−1.38110
$463$	−1.23204e12	−1.24598	−0.622989	−	0.782230i	$-0.714082\pi$
$463$	−1.23204e12	−1.24598	−0.622989	+	0.782230i	$0.714082\pi$
$464$	1.25451e12	1.25645
$465$	0	0
$466$	−2.21988e12	−2.18068
$467$	−1.10796e12	−1.07794	−0.538972	−	0.842323i	$-0.681187\pi$
$467$	−1.10796e12	−1.07794	−0.538972	+	0.842323i	$0.681187\pi$
$468$	1.05049e12	1.01224
$469$	−7.00030e11	−0.668096
$470$	0	0
$471$	6.81444e10	0.0638023
$472$	−3.99033e10	−0.0370058
$473$	−1.78157e12	−1.63654
$474$	1.98304e12	1.80439
$475$	0	0
$476$	1.94343e12	1.73515
$477$	7.23188e11	0.639615
$478$	−7.37249e11	−0.645935
$479$	−6.31846e10	−0.0548405	−0.0274203	−	0.999624i	$-0.508729\pi$
$479$	−6.31846e10	−0.0548405	−0.0274203	+	0.999624i	$0.508729\pi$
$480$	0	0
$481$	−1.05654e12	−0.899978
$482$	−1.75582e12	−1.48173
$483$	−7.01181e11	−0.586230
$484$	2.19060e12	1.81451
$485$	0	0
$486$	1.35050e11	0.109807
$487$	1.15658e12	0.931739	0.465870	−	0.884853i	$-0.345742\pi$
$487$	1.15658e12	0.931739	0.465870	+	0.884853i	$0.345742\pi$
$488$	2.65896e12	2.12237
$489$	−2.36152e10	−0.0186768
$490$	0	0
$491$	1.67660e12	1.30185	0.650927	−	0.759140i	$-0.274380\pi$
$491$	1.67660e12	1.30185	0.650927	+	0.759140i	$0.274380\pi$
$492$	2.92547e11	0.225088
$493$	−1.85769e12	−1.41632
$494$	3.30237e12	2.49491
$495$	0	0
$496$	−1.64616e12	−1.22125
$497$	−1.10039e12	−0.808988
$498$	−9.71892e10	−0.0708086
$499$	−1.70175e12	−1.22869	−0.614347	−	0.789036i	$-0.710581\pi$
$499$	−1.70175e12	−1.22869	−0.614347	+	0.789036i	$0.710581\pi$
$500$	0	0
$501$	2.91581e11	0.206771
$502$	−3.85862e12	−2.71185
$503$	2.10863e12	1.46874	0.734371	−	0.678748i	$-0.237477\pi$
$503$	2.10863e12	1.46874	0.734371	+	0.678748i	$0.237477\pi$
$504$	7.71207e11	0.532395
$505$	0	0
$506$	3.55817e12	2.41295
$507$	−1.26756e12	−0.851984
$508$	−1.35735e11	−0.0904283
$509$	5.19165e11	0.342827	0.171414	−	0.985199i	$-0.445167\pi$
$509$	5.19165e11	0.342827	0.171414	+	0.985199i	$0.445167\pi$
$510$	0	0
$511$	−1.73200e12	−1.12371
$512$	2.25353e12	1.44927
$513$	2.79653e11	0.178275
$514$	4.89924e12	3.09596
$515$	0	0
$516$	2.10830e12	1.30921
$517$	7.08210e11	0.435968
$518$	−1.60967e12	−0.982318
$519$	6.86324e11	0.415218
$520$	0	0
$521$	−1.20828e12	−0.718453	−0.359226	−	0.933251i	$-0.616959\pi$
$521$	−1.20828e12	−0.718453	−0.359226	+	0.933251i	$0.616959\pi$
$522$	−1.52985e12	−0.901842
$523$	−6.58206e10	−0.0384684	−0.0192342	−	0.999815i	$-0.506123\pi$
$523$	−6.58206e10	−0.0384684	−0.0192342	+	0.999815i	$0.506123\pi$
$524$	−3.48373e12	−2.01862
$525$	0	0
$526$	−1.55423e12	−0.885275
$527$	2.43764e12	1.37664
$528$	−1.14166e12	−0.639272
$529$	4.36194e10	0.0242175
$530$	0	0
$531$	1.41955e10	0.00774865
$532$	3.31412e12	1.79376
$533$	−5.92206e11	−0.317834
$534$	−1.24244e11	−0.0661209
$535$	0	0
$536$	−2.02567e12	−1.06005
$537$	−9.35311e11	−0.485369
$538$	−2.00421e12	−1.03139
$539$	1.80697e10	0.00922151
$540$	0	0
$541$	3.28051e12	1.64647	0.823235	−	0.567700i	$-0.192167\pi$
$541$	3.28051e12	1.64647	0.823235	+	0.567700i	$0.192167\pi$
$542$	6.18706e11	0.307955
$543$	−1.77780e12	−0.877576
$544$	−4.23226e11	−0.207194
$545$	0	0
$546$	−3.23982e12	−1.56011
$547$	1.17422e12	0.560798	0.280399	−	0.959884i	$-0.409533\pi$
$547$	1.17422e12	0.560798	0.280399	+	0.959884i	$0.409533\pi$
$548$	3.45875e12	1.63835
$549$	−9.45920e11	−0.444405
$550$	0	0
$551$	−3.16791e12	−1.46416
$552$	−2.02901e12	−0.930160
$553$	−4.02858e12	−1.83184
$554$	1.98106e12	0.893518
$555$	0	0
$556$	4.24994e12	1.88602
$557$	2.35749e12	1.03777	0.518886	−	0.854843i	$-0.326347\pi$
$557$	2.35749e12	1.03777	0.518886	+	0.854843i	$0.326347\pi$
$558$	2.00744e12	0.876576
$559$	−4.26785e12	−1.84866
$560$	0	0
$561$	1.69058e12	0.720614
$562$	−5.17588e11	−0.218862
$563$	1.36679e11	0.0573343	0.0286672	−	0.999589i	$-0.490874\pi$
$563$	1.36679e11	0.0573343	0.0286672	+	0.999589i	$0.490874\pi$
$564$	−8.38091e11	−0.348767
$565$	0	0
$566$	3.97418e12	1.62770
$567$	−2.74356e11	−0.111478
$568$	−3.18419e12	−1.28360
$569$	3.02495e11	0.120980	0.0604899	−	0.998169i	$-0.480734\pi$
$569$	3.02495e11	0.120980	0.0604899	+	0.998169i	$0.480734\pi$
$570$	0	0
$571$	1.76137e12	0.693408	0.346704	−	0.937975i	$-0.387301\pi$
$571$	1.76137e12	0.693408	0.346704	+	0.937975i	$0.387301\pi$
$572$	1.08295e13	4.22985
$573$	−2.68327e12	−1.03985
$574$	−9.02245e11	−0.346913
$575$	0	0
$576$	−1.04855e12	−0.396906
$577$	−3.04220e12	−1.14261	−0.571303	−	0.820739i	$-0.693562\pi$
$577$	−3.04220e12	−1.14261	−0.571303	+	0.820739i	$0.693562\pi$
$578$	9.05061e11	0.337289
$579$	−1.98060e12	−0.732390
$580$	0	0
$581$	1.97441e11	0.0718861
$582$	2.91952e12	1.05477
$583$	7.45533e12	2.67275
$584$	−5.01189e12	−1.78297
$585$	0	0
$586$	−4.06811e12	−1.42513
$587$	−2.00567e12	−0.697249	−0.348624	−	0.937263i	$-0.613351\pi$
$587$	−2.00567e12	−0.697249	−0.348624	+	0.937263i	$0.613351\pi$
$588$	−2.13836e10	−0.00737705
$589$	4.15688e12	1.42314
$590$	0	0
$591$	1.49272e12	0.503310
$592$	−1.35881e12	−0.454687
$593$	5.38900e12	1.78963	0.894813	−	0.446441i	$-0.147309\pi$
$593$	5.38900e12	1.78963	0.894813	+	0.446441i	$0.147309\pi$
$594$	1.39223e12	0.458850
$595$	0	0
$596$	−2.46270e12	−0.799471
$597$	1.76150e12	0.567543
$598$	8.52380e12	2.72570
$599$	8.61267e11	0.273349	0.136674	−	0.990616i	$-0.456359\pi$
$599$	8.61267e11	0.273349	0.136674	+	0.990616i	$0.456359\pi$
$600$	0	0
$601$	1.60177e12	0.500799	0.250400	−	0.968143i	$-0.419438\pi$
$601$	1.60177e12	0.500799	0.250400	+	0.968143i	$0.419438\pi$
$602$	−6.50220e12	−2.01779
$603$	7.20631e11	0.221965
$604$	−2.59857e12	−0.794452
$605$	0	0
$606$	−3.02163e12	−0.910152
$607$	−3.12903e12	−0.935535	−0.467768	−	0.883851i	$-0.654942\pi$
$607$	−3.12903e12	−0.935535	−0.467768	+	0.883851i	$0.654942\pi$
$608$	−7.21724e11	−0.214193
$609$	3.10790e12	0.915565
$610$	0	0
$611$	1.69656e12	0.492474
$612$	−2.00062e12	−0.576479
$613$	3.83573e12	1.09717	0.548587	−	0.836093i	$-0.315166\pi$
$613$	3.83573e12	1.09717	0.548587	+	0.836093i	$0.315166\pi$
$614$	−1.15203e13	−3.27118
$615$	0	0
$616$	7.95035e12	2.22471
$617$	3.16031e12	0.877901	0.438951	−	0.898511i	$-0.355350\pi$
$617$	3.16031e12	0.877901	0.438951	+	0.898511i	$0.355350\pi$
$618$	−5.55614e11	−0.153223
$619$	5.43833e11	0.148887	0.0744437	−	0.997225i	$-0.476282\pi$
$619$	5.43833e11	0.148887	0.0744437	+	0.997225i	$0.476282\pi$
$620$	0	0
$621$	7.21816e11	0.194766
$622$	−8.80697e12	−2.35923
$623$	2.52403e11	0.0671271
$624$	−2.73492e12	−0.722128
$625$	0	0
$626$	5.49930e11	0.143128
$627$	2.88293e12	0.744956
$628$	−8.31333e11	−0.213283
$629$	2.01214e12	0.512543
$630$	0	0
$631$	−3.75043e12	−0.941780	−0.470890	−	0.882192i	$-0.656067\pi$
$631$	−3.75043e12	−0.941780	−0.470890	+	0.882192i	$0.656067\pi$
$632$	−1.16575e13	−2.90655
$633$	−2.57033e12	−0.636316
$634$	3.07878e12	0.756793
$635$	0	0
$636$	−8.82258e12	−2.13815
$637$	4.32871e10	0.0104167
$638$	−1.57711e13	−3.76851
$639$	1.13277e12	0.268774
$640$	0	0
$641$	−1.80612e12	−0.422557	−0.211279	−	0.977426i	$-0.567763\pi$
$641$	−1.80612e12	−0.422557	−0.211279	+	0.977426i	$0.567763\pi$
$642$	1.94638e11	0.0452189
$643$	−2.09654e11	−0.0483674	−0.0241837	−	0.999708i	$-0.507699\pi$
$643$	−2.09654e11	−0.0483674	−0.0241837	+	0.999708i	$0.507699\pi$
$644$	8.55411e12	1.95969
$645$	0	0
$646$	−6.28925e12	−1.42086
$647$	1.95242e12	0.438030	0.219015	−	0.975722i	$-0.429716\pi$
$647$	1.95242e12	0.438030	0.219015	+	0.975722i	$0.429716\pi$
$648$	−7.93902e11	−0.176880
$649$	1.46341e11	0.0323792
$650$	0	0
$651$	−4.07815e12	−0.889915
$652$	2.88095e11	0.0624341
$653$	−1.53186e12	−0.329692	−0.164846	−	0.986319i	$-0.552713\pi$
$653$	−1.53186e12	−0.329692	−0.164846	+	0.986319i	$0.552713\pi$
$654$	−3.08346e12	−0.659080
$655$	0	0
$656$	−7.61638e11	−0.160576
$657$	1.78297e12	0.373336
$658$	2.58476e12	0.537531
$659$	1.91059e12	0.394624	0.197312	−	0.980341i	$-0.436779\pi$
$659$	1.91059e12	0.394624	0.197312	+	0.980341i	$0.436779\pi$
$660$	0	0
$661$	−4.78855e12	−0.975657	−0.487828	−	0.872940i	$-0.662211\pi$
$661$	−4.78855e12	−0.975657	−0.487828	+	0.872940i	$0.662211\pi$
$662$	3.09992e11	0.0627321
$663$	4.04989e12	0.814014
$664$	5.71334e11	0.114060
$665$	0	0
$666$	1.65704e12	0.326361
$667$	−8.17672e12	−1.59961
$668$	−3.55717e12	−0.691210
$669$	3.23026e12	0.623477
$670$	0	0
$671$	−9.75146e12	−1.85703
$672$	7.08053e11	0.133938
$673$	−1.43146e12	−0.268974	−0.134487	−	0.990915i	$-0.542939\pi$
$673$	−1.43146e12	−0.268974	−0.134487	+	0.990915i	$0.542939\pi$
$674$	−1.82218e13	−3.40112
$675$	0	0
$676$	1.54636e13	2.84808
$677$	−4.84298e12	−0.886062	−0.443031	−	0.896506i	$-0.646097\pi$
$677$	−4.84298e12	−0.886062	−0.443031	+	0.896506i	$0.646097\pi$
$678$	7.84199e12	1.42525
$679$	−5.93103e12	−1.07082
$680$	0	0
$681$	−6.62691e11	−0.118073
$682$	2.06947e13	3.66293
$683$	9.33530e12	1.64148	0.820739	−	0.571303i	$-0.193562\pi$
$683$	9.33530e12	1.64148	0.820739	+	0.571303i	$0.193562\pi$
$684$	−3.41164e12	−0.595952
$685$	0	0
$686$	−9.89558e12	−1.70601
$687$	−3.48104e12	−0.596217
$688$	−5.48889e12	−0.933978
$689$	1.78597e13	3.01917
$690$	0	0
$691$	7.11296e12	1.18686	0.593429	−	0.804886i	$-0.297774\pi$
$691$	7.11296e12	1.18686	0.593429	+	0.804886i	$0.297774\pi$
$692$	−8.37286e12	−1.38802
$693$	−2.82833e12	−0.465833
$694$	1.19278e13	1.95184
$695$	0	0
$696$	8.99332e12	1.45271
$697$	1.12784e12	0.181008
$698$	−2.03358e13	−3.24273
$699$	−4.64242e12	−0.735525
$700$	0	0
$701$	9.01666e12	1.41031	0.705155	−	0.709053i	$-0.250877\pi$
$701$	9.01666e12	1.41031	0.705155	+	0.709053i	$0.250877\pi$
$702$	3.33516e12	0.518322
$703$	3.43128e12	0.529856
$704$	−1.08095e13	−1.65855
$705$	0	0
$706$	−4.66129e12	−0.706130
$707$	6.13847e12	0.924002
$708$	−1.73179e11	−0.0259028
$709$	−1.21803e13	−1.81029	−0.905146	−	0.425101i	$-0.860239\pi$
$709$	−1.21803e13	−1.81029	−0.905146	+	0.425101i	$0.860239\pi$
$710$	0	0
$711$	4.14713e12	0.608604
$712$	7.30377e11	0.106509
$713$	1.07294e13	1.55479
$714$	6.17013e12	0.888489
$715$	0	0
$716$	1.14104e13	1.62253
$717$	−1.54181e12	−0.217868
$718$	−6.96607e11	−0.0978200
$719$	9.77884e12	1.36461	0.682303	−	0.731069i	$-0.260978\pi$
$719$	9.77884e12	1.36461	0.682303	+	0.731069i	$0.260978\pi$
$720$	0	0
$721$	1.12874e12	0.155555
$722$	1.77331e12	0.242867
$723$	−3.67194e12	−0.499773
$724$	2.16884e13	2.93363
$725$	0	0
$726$	6.95486e12	0.929124
$727$	−9.28712e12	−1.23304	−0.616519	−	0.787340i	$-0.711458\pi$
$727$	−9.28712e12	−1.23304	−0.616519	+	0.787340i	$0.711458\pi$
$728$	1.90455e13	2.51306
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	8.12798e12	1.05282
$732$	1.15398e13	1.48559
$733$	6.00986e12	0.768947	0.384473	−	0.923136i	$-0.374383\pi$
$733$	6.00986e12	0.768947	0.384473	+	0.923136i	$0.374383\pi$
$734$	1.77214e13	2.25354
$735$	0	0
$736$	−1.86285e12	−0.234007
$737$	7.42896e12	0.927523
$738$	9.28796e11	0.115257
$739$	−2.67574e12	−0.330023	−0.165011	−	0.986292i	$-0.552766\pi$
$739$	−2.67574e12	−0.330023	−0.165011	+	0.986292i	$0.552766\pi$
$740$	0	0
$741$	6.90624e12	0.841511
$742$	2.72098e13	3.29539
$743$	−1.25716e12	−0.151335	−0.0756676	−	0.997133i	$-0.524109\pi$
$743$	−1.25716e12	−0.151335	−0.0756676	+	0.997133i	$0.524109\pi$
$744$	−1.18009e13	−1.41201
$745$	0	0
$746$	−1.66162e13	−1.96429
$747$	−2.03251e11	−0.0238831
$748$	−2.06244e13	−2.40892
$749$	−3.95410e11	−0.0459071
$750$	0	0
$751$	−8.47897e10	−0.00972665	−0.00486332	−	0.999988i	$-0.501548\pi$
$751$	−8.47897e10	−0.00972665	−0.00486332	+	0.999988i	$0.501548\pi$
$752$	2.18195e12	0.248808
$753$	−8.06951e12	−0.914681
$754$	−3.77807e13	−4.25695
$755$	0	0
$756$	3.34702e12	0.372658
$757$	7.14193e12	0.790467	0.395234	−	0.918581i	$-0.370664\pi$
$757$	7.14193e12	0.790467	0.395234	+	0.918581i	$0.370664\pi$
$758$	1.15211e13	1.26760
$759$	7.44118e12	0.813868
$760$	0	0
$761$	6.52955e12	0.705753	0.352876	−	0.935670i	$-0.385204\pi$
$761$	6.52955e12	0.705753	0.352876	+	0.935670i	$0.385204\pi$
$762$	−4.30939e11	−0.0463040
$763$	6.26408e12	0.669109
$764$	3.27348e13	3.47608
$765$	0	0
$766$	−1.45913e13	−1.53131
$767$	3.50569e11	0.0365759
$768$	1.05888e13	1.09830
$769$	−8.49134e12	−0.875604	−0.437802	−	0.899071i	$-0.644243\pi$
$769$	−8.49134e12	−0.875604	−0.437802	+	0.899071i	$0.644243\pi$
$770$	0	0
$771$	1.02458e13	1.04424
$772$	2.41624e13	2.44829
$773$	−1.20096e13	−1.20982	−0.604912	−	0.796292i	$-0.706792\pi$
$773$	−1.20096e13	−1.20982	−0.604912	+	0.796292i	$0.706792\pi$
$774$	6.69355e12	0.670382
$775$	0	0
$776$	−1.71626e13	−1.69905
$777$	−3.36629e12	−0.331327
$778$	−1.78397e13	−1.74574
$779$	1.92329e12	0.187123
$780$	0	0
$781$	1.16777e13	1.12312
$782$	−1.62333e13	−1.55230
$783$	−3.19936e12	−0.304183
$784$	5.56716e10	0.00526273
$785$	0	0
$786$	−1.10604e13	−1.03364
$787$	−1.46010e13	−1.35673	−0.678367	−	0.734723i	$-0.737312\pi$
$787$	−1.46010e13	−1.35673	−0.678367	+	0.734723i	$0.737312\pi$
$788$	−1.82106e13	−1.68250
$789$	−3.25034e12	−0.298595
$790$	0	0
$791$	−1.59311e13	−1.44694
$792$	−8.18432e12	−0.739127
$793$	−2.33602e13	−2.09772
$794$	−5.95678e12	−0.531887
$795$	0	0
$796$	−2.14896e13	−1.89722
$797$	−1.09073e13	−0.957533	−0.478767	−	0.877942i	$-0.658916\pi$
$797$	−1.09073e13	−0.957533	−0.478767	+	0.877942i	$0.658916\pi$
$798$	1.05219e13	0.918501
$799$	−3.23104e12	−0.280467
$800$	0	0
$801$	−2.59830e11	−0.0223020
$802$	−9.66297e12	−0.824757
$803$	1.83806e13	1.56006
$804$	−8.79138e12	−0.742002
$805$	0	0
$806$	4.95753e13	4.13769
$807$	−4.19140e12	−0.347879
$808$	1.77629e13	1.46609
$809$	−4.54621e12	−0.373148	−0.186574	−	0.982441i	$-0.559738\pi$
$809$	−4.54621e12	−0.373148	−0.186574	+	0.982441i	$0.559738\pi$
$810$	0	0
$811$	−2.32325e13	−1.88583	−0.942916	−	0.333031i	$-0.891928\pi$
$811$	−2.32325e13	−1.88583	−0.942916	+	0.333031i	$0.891928\pi$
$812$	−3.79150e13	−3.06062
$813$	1.29390e12	0.103871
$814$	1.70823e13	1.36376
$815$	0	0
$816$	5.20856e12	0.411256
$817$	1.38606e13	1.08838
$818$	−1.50755e13	−1.17728
$819$	−6.77542e12	−0.526210
$820$	0	0
$821$	−5.51930e12	−0.423975	−0.211987	−	0.977272i	$-0.567994\pi$
$821$	−5.51930e12	−0.423975	−0.211987	+	0.977272i	$0.567994\pi$
$822$	1.09811e13	0.838921
$823$	−2.84213e11	−0.0215946	−0.0107973	−	0.999942i	$-0.503437\pi$
$823$	−2.84213e11	−0.0215946	−0.0107973	+	0.999942i	$0.503437\pi$
$824$	3.26622e12	0.246816
$825$	0	0
$826$	5.34103e11	0.0399222
$827$	−1.01097e13	−0.751562	−0.375781	−	0.926708i	$-0.622626\pi$
$827$	−1.01097e13	−0.751562	−0.375781	+	0.926708i	$0.622626\pi$
$828$	−8.80584e12	−0.651080
$829$	1.66445e13	1.22398	0.611992	−	0.790864i	$-0.290369\pi$
$829$	1.66445e13	1.22398	0.611992	+	0.790864i	$0.290369\pi$
$830$	0	0
$831$	4.14298e12	0.301376
$832$	−2.58947e13	−1.87351
$833$	−8.24388e10	−0.00593238
$834$	1.34930e13	0.965742
$835$	0	0
$836$	−3.51705e13	−2.49029
$837$	4.19816e12	0.295661
$838$	1.04839e13	0.734384
$839$	−2.76169e13	−1.92418	−0.962090	−	0.272731i	$-0.912073\pi$
$839$	−2.76169e13	−1.92418	−0.962090	+	0.272731i	$0.912073\pi$
$840$	0	0
$841$	2.17352e13	1.49824
$842$	−9.27706e11	−0.0636071
$843$	−1.08243e12	−0.0738203
$844$	3.13570e13	2.12713
$845$	0	0
$846$	−2.66082e12	−0.178587
$847$	−1.41289e13	−0.943263
$848$	2.29694e13	1.52534
$849$	8.31118e12	0.549007
$850$	0	0
$851$	8.85654e12	0.578870
$852$	−1.38193e13	−0.898479
$853$	3.79166e12	0.245222	0.122611	−	0.992455i	$-0.460873\pi$
$853$	3.79166e12	0.245222	0.122611	+	0.992455i	$0.460873\pi$
$854$	−3.55900e13	−2.28964
$855$	0	0
$856$	−1.14420e12	−0.0728398
$857$	7.65543e12	0.484792	0.242396	−	0.970177i	$-0.422067\pi$
$857$	7.65543e12	0.484792	0.242396	+	0.970177i	$0.422067\pi$
$858$	3.43821e13	2.16591
$859$	−1.96425e13	−1.23091	−0.615456	−	0.788171i	$-0.711028\pi$
$859$	−1.96425e13	−1.23091	−0.615456	+	0.788171i	$0.711028\pi$
$860$	0	0
$861$	−1.88686e12	−0.117011
$862$	1.56017e13	0.962475
$863$	−2.18218e13	−1.33919	−0.669595	−	0.742726i	$-0.733532\pi$
$863$	−2.18218e13	−1.33919	−0.669595	+	0.742726i	$0.733532\pi$
$864$	−7.28890e11	−0.0444990
$865$	0	0
$866$	1.29807e13	0.784276
$867$	1.89275e12	0.113765
$868$	4.97516e13	2.97487
$869$	4.27527e13	2.54316
$870$	0	0
$871$	1.77965e13	1.04774
$872$	1.81263e13	1.06166
$873$	6.10557e12	0.355764
$874$	−2.76825e13	−1.60474
$875$	0	0
$876$	−2.17515e13	−1.24802
$877$	3.16753e12	0.180810	0.0904050	−	0.995905i	$-0.471184\pi$
$877$	3.16753e12	0.180810	0.0904050	+	0.995905i	$0.471184\pi$
$878$	−2.71289e13	−1.54066
$879$	−8.50763e12	−0.480683
$880$	0	0
$881$	1.53485e13	0.858367	0.429184	−	0.903217i	$-0.358801\pi$
$881$	1.53485e13	0.858367	0.429184	+	0.903217i	$0.358801\pi$
$882$	−6.78900e10	−0.00377744
$883$	−2.12370e13	−1.17563	−0.587815	−	0.808996i	$-0.700012\pi$
$883$	−2.12370e13	−1.17563	−0.587815	+	0.808996i	$0.700012\pi$
$884$	−4.94068e13	−2.72115
$885$	0	0
$886$	−5.73576e12	−0.312708
$887$	1.92895e13	1.04632	0.523161	−	0.852234i	$-0.324753\pi$
$887$	1.92895e13	1.04632	0.523161	+	0.852234i	$0.324753\pi$
$888$	−9.74102e12	−0.525710
$889$	8.75459e11	0.0470087
$890$	0	0
$891$	2.91156e12	0.154766
$892$	−3.94078e13	−2.08421
$893$	−5.50986e12	−0.289941
$894$	−7.81873e12	−0.409371
$895$	0	0
$896$	−3.49758e13	−1.81293
$897$	1.78258e13	0.919354
$898$	3.08940e13	1.58537
$899$	−4.75567e13	−2.42825
$900$	0	0
$901$	−3.40131e13	−1.71943
$902$	9.57493e12	0.481622
$903$	−1.35980e13	−0.680583
$904$	−4.60997e13	−2.29583
$905$	0	0
$906$	−8.25010e12	−0.406801
$907$	6.79922e12	0.333600	0.166800	−	0.985991i	$-0.446657\pi$
$907$	6.79922e12	0.333600	0.166800	+	0.985991i	$0.446657\pi$
$908$	8.08455e12	0.394702
$909$	−6.31911e12	−0.306986
$910$	0	0
$911$	−1.69638e13	−0.816002	−0.408001	−	0.912981i	$-0.633774\pi$
$911$	−1.69638e13	−0.816002	−0.408001	+	0.912981i	$0.633774\pi$
$912$	8.88212e12	0.425148
$913$	−2.09531e12	−0.0998000
$914$	3.97162e13	1.88239
$915$	0	0
$916$	4.24672e13	1.99308
$917$	2.24693e13	1.04937
$918$	−6.35170e12	−0.295187
$919$	−2.00840e13	−0.928818	−0.464409	−	0.885621i	$-0.653733\pi$
$919$	−2.00840e13	−0.928818	−0.464409	+	0.885621i	$0.653733\pi$
$920$	0	0
$921$	−2.40923e13	−1.10334
$922$	2.60182e13	1.18574
$923$	2.79746e13	1.26869
$924$	3.45044e13	1.55722
$925$	0	0
$926$	4.77193e13	2.13277
$927$	−1.16195e12	−0.0516809
$928$	8.25686e12	0.365468
$929$	2.08690e13	0.919246	0.459623	−	0.888114i	$-0.347985\pi$
$929$	2.08690e13	0.919246	0.459623	+	0.888114i	$0.347985\pi$
$930$	0	0
$931$	−1.40582e11	−0.00613277
$932$	5.66355e13	2.45877
$933$	−1.84180e13	−0.795747
$934$	4.29133e13	1.84515
$935$	0	0
$936$	−1.96060e13	−0.834926
$937$	−5.34149e12	−0.226378	−0.113189	−	0.993573i	$-0.536107\pi$
$937$	−5.34149e12	−0.226378	−0.113189	+	0.993573i	$0.536107\pi$
$938$	2.71135e13	1.14360
$939$	1.15007e12	0.0482756
$940$	0	0
$941$	−5.49799e12	−0.228587	−0.114293	−	0.993447i	$-0.536460\pi$
$941$	−5.49799e12	−0.228587	−0.114293	+	0.993447i	$0.536460\pi$
$942$	−2.63937e12	−0.109212
$943$	4.96423e12	0.204432
$944$	4.50868e11	0.0184789
$945$	0	0
$946$	6.90036e13	2.80131
$947$	−3.05434e13	−1.23408	−0.617038	−	0.786933i	$-0.711668\pi$
$947$	−3.05434e13	−1.23408	−0.617038	+	0.786933i	$0.711668\pi$
$948$	−5.05932e13	−2.03449
$949$	4.40318e13	1.76226
$950$	0	0
$951$	6.43864e12	0.255259
$952$	−3.62716e13	−1.43120
$953$	6.24008e12	0.245060	0.122530	−	0.992465i	$-0.460899\pi$
$953$	6.24008e12	0.245060	0.122530	+	0.992465i	$0.460899\pi$
$954$	−2.80105e13	−1.09485
$955$	0	0
$956$	1.88094e13	0.728306
$957$	−3.29821e13	−1.27109
$958$	2.44726e12	0.0938720
$959$	−2.23082e13	−0.851687
$960$	0	0
$961$	3.59637e13	1.36022
$962$	4.09218e13	1.54052
$963$	4.07046e11	0.0152519
$964$	4.47960e13	1.67068
$965$	0	0
$966$	2.71581e13	1.00347
$967$	−4.92222e12	−0.181026	−0.0905131	−	0.995895i	$-0.528851\pi$
$967$	−4.92222e12	−0.181026	−0.0905131	+	0.995895i	$0.528851\pi$
$968$	−4.08847e13	−1.49666
$969$	−1.31527e13	−0.479245
$970$	0	0
$971$	−4.52224e13	−1.63255	−0.816277	−	0.577661i	$-0.803966\pi$
$971$	−4.52224e13	−1.63255	−0.816277	+	0.577661i	$0.803966\pi$
$972$	−3.44552e12	−0.123810
$973$	−2.74112e13	−0.980438
$974$	−4.47965e13	−1.59488
$975$	0	0
$976$	−3.00436e13	−1.05981
$977$	1.93987e13	0.681157	0.340578	−	0.940216i	$-0.389377\pi$
$977$	1.93987e13	0.681157	0.340578	+	0.940216i	$0.389377\pi$
$978$	9.14664e11	0.0319696
$979$	−2.67858e12	−0.0931930
$980$	0	0
$981$	−6.44842e12	−0.222302
$982$	−6.49379e13	−2.22842
$983$	3.77977e13	1.29114	0.645571	−	0.763700i	$-0.276619\pi$
$983$	3.77977e13	1.29114	0.645571	+	0.763700i	$0.276619\pi$
$984$	−5.46000e12	−0.185659
$985$	0	0
$986$	7.19520e13	2.42436
$987$	5.40550e12	0.181305
$988$	−8.42531e13	−2.81306
$989$	3.57757e13	1.18906
$990$	0	0
$991$	3.90411e13	1.28585	0.642925	−	0.765929i	$-0.277721\pi$
$991$	3.90411e13	1.28585	0.642925	+	0.765929i	$0.277721\pi$
$992$	−1.08345e13	−0.355229
$993$	6.48285e11	0.0211590
$994$	4.26202e13	1.38477
$995$	0	0
$996$	2.47958e12	0.0798382
$997$	7.10765e12	0.227823	0.113912	−	0.993491i	$-0.463662\pi$
$997$	7.10765e12	0.227823	0.113912	+	0.993491i	$0.463662\pi$
$998$	6.59122e13	2.10319
$999$	3.46535e12	0.110079

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.10.a.k.1.1	yes	4
3.2	odd	2		225.10.a.r.1.4		4
5.2	odd	4		75.10.b.h.49.2		8
5.3	odd	4		75.10.b.h.49.7		8
5.4	even	2		75.10.a.j.1.4	✓	4
15.2	even	4		225.10.b.n.199.7		8
15.8	even	4		225.10.b.n.199.2		8
15.14	odd	2		225.10.a.t.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.10.a.j.1.4	✓	4	5.4	even	2
75.10.a.k.1.1	yes	4	1.1	even	1	trivial
75.10.b.h.49.2		8	5.2	odd	4
75.10.b.h.49.7		8	5.3	odd	4
225.10.a.r.1.4		4	3.2	odd	2
225.10.a.t.1.1		4	15.14	odd	2
225.10.b.n.199.2		8	15.8	even	4
225.10.b.n.199.7		8	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q-38.7320 q^{2} -81.0000 q^{3} +988.165 q^{4} +3137.29 q^{6} -6373.44 q^{7} -18442.8 q^{8} +6561.00 q^{9} +O(q^{10})\)	\(q-38.7320 q^{2} -81.0000 q^{3} +988.165 q^{4} +3137.29 q^{6} -6373.44 q^{7} -18442.8 q^{8} +6561.00 q^{9} +67637.2 q^{11} -80041.4 q^{12} +162029. q^{13} +246856. q^{14} +208386. q^{16} -308578. q^{17} -254120. q^{18} -526216. q^{19} +516249. q^{21} -2.61972e6 q^{22} -1.35822e6 q^{23} +1.49387e6 q^{24} -6.27569e6 q^{26} -531441. q^{27} -6.29801e6 q^{28} +6.02016e6 q^{29} -7.89958e6 q^{31} +1.37153e6 q^{32} -5.47861e6 q^{33} +1.19518e7 q^{34} +6.48335e6 q^{36} -6.52067e6 q^{37} +2.03814e7 q^{38} -1.31243e7 q^{39} -3.65494e6 q^{41} -1.99953e7 q^{42} -2.63401e7 q^{43} +6.68367e7 q^{44} +5.26067e7 q^{46} +1.04707e7 q^{47} -1.68792e7 q^{48} +267157. q^{49} +2.49948e7 q^{51} +1.60111e8 q^{52} +1.10225e8 q^{53} +2.05838e7 q^{54} +1.17544e8 q^{56} +4.26235e7 q^{57} -2.33173e8 q^{58} +2.16362e6 q^{59} -1.44173e8 q^{61} +3.05966e8 q^{62} -4.18162e7 q^{63} -1.59816e8 q^{64} +2.12197e8 q^{66} +1.09836e8 q^{67} -3.04926e8 q^{68} +1.10016e8 q^{69} +1.72652e8 q^{71} -1.21003e8 q^{72} +2.71753e8 q^{73} +2.52559e8 q^{74} -5.19988e8 q^{76} -4.31082e8 q^{77} +5.08331e8 q^{78} +6.32088e8 q^{79} +4.30467e7 q^{81} +1.41563e8 q^{82} -3.09787e7 q^{83} +5.10139e8 q^{84} +1.02020e9 q^{86} -4.87633e8 q^{87} -1.24742e9 q^{88} -3.96023e7 q^{89} -1.03268e9 q^{91} -1.34215e9 q^{92} +6.39866e8 q^{93} -4.05552e8 q^{94} -1.11094e8 q^{96} +9.30586e8 q^{97} -1.03475e7 q^{98} +4.43767e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 324 q^{3} + 1792 q^{4} - 162 q^{6} - 13036 q^{7} + 24636 q^{8} + 26244 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 - 324 * q^3 + 1792 * q^4 - 162 * q^6 - 13036 * q^7 + 24636 * q^8 + 26244 * q^9	\( 4 q + 2 q^{2} - 324 q^{3} + 1792 q^{4} - 162 q^{6} - 13036 q^{7} + 24636 q^{8} + 26244 q^{9} + 104696 q^{11} - 145152 q^{12} - 140812 q^{13} - 181062 q^{14} + 1319800 q^{16} - 489352 q^{17} + 13122 q^{18} + 393308 q^{19} + 1055916 q^{21} + 793956 q^{22} - 2326488 q^{23} - 1995516 q^{24} - 6477502 q^{26} - 2125764 q^{27} - 10094272 q^{28} + 4926616 q^{29} - 97516 q^{31} + 29228344 q^{32} - 8480376 q^{33} + 16828644 q^{34} + 11757312 q^{36} + 4958984 q^{37} + 43844342 q^{38} + 11405772 q^{39} - 996656 q^{41} + 14666022 q^{42} - 28298860 q^{43} + 206545216 q^{44} + 103917804 q^{46} + 30714920 q^{47} - 106903800 q^{48} + 80698920 q^{49} + 39637512 q^{51} + 146870528 q^{52} - 70694368 q^{53} - 1062882 q^{54} + 170654220 q^{56} - 31857948 q^{57} - 532031436 q^{58} + 225946712 q^{59} + 6295340 q^{61} + 665342778 q^{62} - 85529196 q^{63} + 632883328 q^{64} - 64310436 q^{66} + 217434788 q^{67} - 603488576 q^{68} + 188445528 q^{69} + 22716688 q^{71} + 161636796 q^{72} - 79864888 q^{73} + 193190788 q^{74} + 641703872 q^{76} - 395948232 q^{77} + 524677662 q^{78} + 1276962080 q^{79} + 172186884 q^{81} + 186865488 q^{82} - 175482984 q^{83} + 817636032 q^{84} + 1426174466 q^{86} - 399055896 q^{87} + 2581163256 q^{88} - 897754752 q^{89} - 1837903676 q^{91} + 1998210624 q^{92} + 7898796 q^{93} - 2190567228 q^{94} - 2367495864 q^{96} - 1702783612 q^{97} - 591217148 q^{98} + 686910456 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 324 * q^3 + 1792 * q^4 - 162 * q^6 - 13036 * q^7 + 24636 * q^8 + 26244 * q^9 + 104696 * q^11 - 145152 * q^12 - 140812 * q^13 - 181062 * q^14 + 1319800 * q^16 - 489352 * q^17 + 13122 * q^18 + 393308 * q^19 + 1055916 * q^21 + 793956 * q^22 - 2326488 * q^23 - 1995516 * q^24 - 6477502 * q^26 - 2125764 * q^27 - 10094272 * q^28 + 4926616 * q^29 - 97516 * q^31 + 29228344 * q^32 - 8480376 * q^33 + 16828644 * q^34 + 11757312 * q^36 + 4958984 * q^37 + 43844342 * q^38 + 11405772 * q^39 - 996656 * q^41 + 14666022 * q^42 - 28298860 * q^43 + 206545216 * q^44 + 103917804 * q^46 + 30714920 * q^47 - 106903800 * q^48 + 80698920 * q^49 + 39637512 * q^51 + 146870528 * q^52 - 70694368 * q^53 - 1062882 * q^54 + 170654220 * q^56 - 31857948 * q^57 - 532031436 * q^58 + 225946712 * q^59 + 6295340 * q^61 + 665342778 * q^62 - 85529196 * q^63 + 632883328 * q^64 - 64310436 * q^66 + 217434788 * q^67 - 603488576 * q^68 + 188445528 * q^69 + 22716688 * q^71 + 161636796 * q^72 - 79864888 * q^73 + 193190788 * q^74 + 641703872 * q^76 - 395948232 * q^77 + 524677662 * q^78 + 1276962080 * q^79 + 172186884 * q^81 + 186865488 * q^82 - 175482984 * q^83 + 817636032 * q^84 + 1426174466 * q^86 - 399055896 * q^87 + 2581163256 * q^88 - 897754752 * q^89 - 1837903676 * q^91 + 1998210624 * q^92 + 7898796 * q^93 - 2190567228 * q^94 - 2367495864 * q^96 - 1702783612 * q^97 - 591217148 * q^98 + 686910456 * q^99

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