LMFDB - Embedded newform 75.10.a.j.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			$33.1381$ 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)

$f(q)$	$=$	$q-44.2736 q^{2} +81.0000 q^{3} +1448.15 q^{4} -3586.16 q^{6} +3879.20 q^{7} -41447.0 q^{8} +6561.00 q^{9} +O(q^{10})$	\(q-44.2736 q^{2} +81.0000 q^{3} +1448.15 q^{4} -3586.16 q^{6} +3879.20 q^{7} -41447.0 q^{8} +6561.00 q^{9} +84809.6 q^{11} +117301. q^{12} +58662.8 q^{13} -171746. q^{14} +1.09355e6 q^{16} +176199. q^{17} -290479. q^{18} +800041. q^{19} +314215. q^{21} -3.75483e6 q^{22} -1.65723e6 q^{23} -3.35720e6 q^{24} -2.59721e6 q^{26} +531441. q^{27} +5.61768e6 q^{28} -4.70073e6 q^{29} +5.05198e6 q^{31} -2.71947e7 q^{32} +6.86958e6 q^{33} -7.80099e6 q^{34} +9.50134e6 q^{36} +4.68753e6 q^{37} -3.54207e7 q^{38} +4.75168e6 q^{39} -4.71307e6 q^{41} -1.39114e7 q^{42} -1.65548e7 q^{43} +1.22817e8 q^{44} +7.33718e7 q^{46} +1.79860e7 q^{47} +8.85778e7 q^{48} -2.53054e7 q^{49} +1.42722e7 q^{51} +8.49527e7 q^{52} +1.50727e7 q^{53} -2.35288e7 q^{54} -1.60781e8 q^{56} +6.48033e7 q^{57} +2.08118e8 q^{58} +1.15379e7 q^{59} -3.48684e7 q^{61} -2.23669e8 q^{62} +2.54514e7 q^{63} +6.44110e8 q^{64} -3.04141e8 q^{66} +1.51756e7 q^{67} +2.55164e8 q^{68} -1.34236e8 q^{69} -3.98346e7 q^{71} -2.71934e8 q^{72} -7.02902e7 q^{73} -2.07534e8 q^{74} +1.15858e9 q^{76} +3.28993e8 q^{77} -2.10374e8 q^{78} +1.16781e8 q^{79} +4.30467e7 q^{81} +2.08665e8 q^{82} +7.69619e8 q^{83} +4.55032e8 q^{84} +7.32941e8 q^{86} -3.80759e8 q^{87} -3.51510e9 q^{88} -9.57076e8 q^{89} +2.27564e8 q^{91} -2.39993e9 q^{92} +4.09210e8 q^{93} -7.96305e8 q^{94} -2.20277e9 q^{96} +4.93418e8 q^{97} +1.12036e9 q^{98} +5.56436e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−44.2736	−1.95664	−0.978318	−	0.207107i	$-0.933595\pi$
$2$	−44.2736	−1.95664	−0.978318	+	0.207107i	$0.933595\pi$
$3$	81.0000	0.577350
$3$	81.0000	0.577350
$4$	1448.15	2.82843
$5$	0	0
$5$	0	0
$6$	−3586.16	−1.12966
$7$	3879.20	0.610662	0.305331	−	0.952246i	$-0.401233\pi$
$7$	3879.20	0.610662	0.305331	+	0.952246i	$0.401233\pi$
$8$	−41447.0	−3.57757
$9$	6561.00	0.333333
$10$	0	0
$11$	84809.6	1.74654	0.873269	−	0.487239i	$-0.161996\pi$
$11$	84809.6	1.74654	0.873269	+	0.487239i	$0.161996\pi$
$12$	117301.	1.63299
$13$	58662.8	0.569662	0.284831	−	0.958578i	$-0.408062\pi$
$13$	58662.8	0.569662	0.284831	+	0.958578i	$0.408062\pi$
$14$	−171746.	−1.19484
$15$	0	0
$16$	1.09355e6	4.17157
$17$	176199.	0.511663	0.255832	−	0.966721i	$-0.417651\pi$
$17$	176199.	0.511663	0.255832	+	0.966721i	$0.417651\pi$
$18$	−290479.	−0.652212
$19$	800041.	1.40838	0.704192	−	0.710009i	$-0.251309\pi$
$19$	800041.	1.40838	0.704192	+	0.710009i	$0.251309\pi$
$20$	0	0
$21$	314215.	0.352566
$22$	−3.75483e6	−3.41734
$23$	−1.65723e6	−1.23483	−0.617417	−	0.786636i	$-0.711821\pi$
$23$	−1.65723e6	−1.23483	−0.617417	+	0.786636i	$0.711821\pi$
$24$	−3.35720e6	−2.06551
$25$	0	0
$26$	−2.59721e6	−1.11462
$27$	531441.	0.192450
$28$	5.61768e6	1.72721
$29$	−4.70073e6	−1.23417	−0.617084	−	0.786897i	$-0.711686\pi$
$29$	−4.70073e6	−1.23417	−0.617084	+	0.786897i	$0.711686\pi$
$30$	0	0
$31$	5.05198e6	0.982503	0.491251	−	0.871018i	$-0.336540\pi$
$31$	5.05198e6	0.982503	0.491251	+	0.871018i	$0.336540\pi$
$32$	−2.71947e7	−4.58469
$33$	6.86958e6	1.00836
$34$	−7.80099e6	−1.00114
$35$	0	0
$36$	9.50134e6	0.942809
$37$	4.68753e6	0.411184	0.205592	−	0.978638i	$-0.434088\pi$
$37$	4.68753e6	0.411184	0.205592	+	0.978638i	$0.434088\pi$
$38$	−3.54207e7	−2.75570
$39$	4.75168e6	0.328895
$40$	0	0
$41$	−4.71307e6	−0.260481	−0.130241	−	0.991482i	$-0.541575\pi$
$41$	−4.71307e6	−0.260481	−0.130241	+	0.991482i	$0.541575\pi$
$42$	−1.39114e7	−0.689843
$43$	−1.65548e7	−0.738441	−0.369221	−	0.929342i	$-0.620375\pi$
$43$	−1.65548e7	−0.738441	−0.369221	+	0.929342i	$0.620375\pi$
$44$	1.22817e8	4.93995
$45$	0	0
$46$	7.33718e7	2.41612
$47$	1.79860e7	0.537643	0.268821	−	0.963190i	$-0.413366\pi$
$47$	1.79860e7	0.537643	0.268821	+	0.963190i	$0.413366\pi$
$48$	8.85778e7	2.40846
$49$	−2.53054e7	−0.627092
$50$	0	0
$51$	1.42722e7	0.295409
$52$	8.49527e7	1.61125
$53$	1.50727e7	0.262391	0.131195	−	0.991357i	$-0.458118\pi$
$53$	1.50727e7	0.262391	0.131195	+	0.991357i	$0.458118\pi$
$54$	−2.35288e7	−0.376555
$55$	0	0
$56$	−1.60781e8	−2.18468
$57$	6.48033e7	0.813131
$58$	2.08118e8	2.41482
$59$	1.15379e7	0.123963	0.0619814	−	0.998077i	$-0.480258\pi$
$59$	1.15379e7	0.123963	0.0619814	+	0.998077i	$0.480258\pi$
$60$	0	0
$61$	−3.48684e7	−0.322439	−0.161219	−	0.986919i	$-0.551543\pi$
$61$	−3.48684e7	−0.322439	−0.161219	+	0.986919i	$0.551543\pi$
$62$	−2.23669e8	−1.92240
$63$	2.54514e7	0.203554
$64$	6.44110e8	4.79899
$65$	0	0
$66$	−3.04141e8	−1.97300
$67$	1.51756e7	0.0920044	0.0460022	−	0.998941i	$-0.485352\pi$
$67$	1.51756e7	0.0920044	0.0460022	+	0.998941i	$0.485352\pi$
$68$	2.55164e8	1.44720
$69$	−1.34236e8	−0.712932
$70$	0	0
$71$	−3.98346e7	−0.186036	−0.0930182	−	0.995664i	$-0.529651\pi$
$71$	−3.98346e7	−0.186036	−0.0930182	+	0.995664i	$0.529651\pi$
$72$	−2.71934e8	−1.19252
$73$	−7.02902e7	−0.289696	−0.144848	−	0.989454i	$-0.546269\pi$
$73$	−7.02902e7	−0.289696	−0.144848	+	0.989454i	$0.546269\pi$
$74$	−2.07534e8	−0.804538
$75$	0	0
$76$	1.15858e9	3.98351
$77$	3.28993e8	1.06654
$78$	−2.10374e8	−0.643527
$79$	1.16781e8	0.337327	0.168663	−	0.985674i	$-0.446055\pi$
$79$	1.16781e8	0.337327	0.168663	+	0.985674i	$0.446055\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	2.08665e8	0.509667
$83$	7.69619e8	1.78002	0.890009	−	0.455943i	$-0.150698\pi$
$83$	7.69619e8	1.78002	0.890009	+	0.455943i	$0.150698\pi$
$84$	4.55032e8	0.997207
$85$	0	0
$86$	7.32941e8	1.44486
$87$	−3.80759e8	−0.712547
$88$	−3.51510e9	−6.24836
$89$	−9.57076e8	−1.61693	−0.808466	−	0.588543i	$-0.799702\pi$
$89$	−9.57076e8	−1.61693	−0.808466	+	0.588543i	$0.799702\pi$
$90$	0	0
$91$	2.27564e8	0.347871
$92$	−2.39993e9	−3.49264
$93$	4.09210e8	0.567248
$94$	−7.96305e8	−1.05197
$95$	0	0
$96$	−2.20277e9	−2.64697
$97$	4.93418e8	0.565903	0.282952	−	0.959134i	$-0.408686\pi$
$97$	4.93418e8	0.565903	0.282952	+	0.959134i	$0.408686\pi$
$98$	1.12036e9	1.22699
$99$	5.56436e8	0.582179
$100$	0	0
$101$	7.99037e8	0.764048	0.382024	−	0.924152i	$-0.375227\pi$
$101$	7.99037e8	0.764048	0.382024	+	0.924152i	$0.375227\pi$
$102$	−6.31880e8	−0.578008
$103$	2.08570e9	1.82593	0.912966	−	0.408036i	$-0.133786\pi$
$103$	2.08570e9	1.82593	0.912966	+	0.408036i	$0.133786\pi$
$104$	−2.43139e9	−2.03800
$105$	0	0
$106$	−6.67321e8	−0.513403
$107$	1.16423e9	0.858640	0.429320	−	0.903152i	$-0.358753\pi$
$107$	1.16423e9	0.858640	0.429320	+	0.903152i	$0.358753\pi$
$108$	7.69609e8	0.544331
$109$	−4.41131e8	−0.299329	−0.149664	−	0.988737i	$-0.547819\pi$
$109$	−4.41131e8	−0.299329	−0.149664	+	0.988737i	$0.547819\pi$
$110$	0	0
$111$	3.79690e8	0.237397
$112$	4.24211e9	2.54742
$113$	−2.92946e8	−0.169019	−0.0845094	−	0.996423i	$-0.526932\pi$
$113$	−2.92946e8	−0.169019	−0.0845094	+	0.996423i	$0.526932\pi$
$114$	−2.86908e9	−1.59100
$115$	0	0
$116$	−6.80738e9	−3.49075
$117$	3.84886e8	0.189887
$118$	−5.10823e8	−0.242550
$119$	6.83513e8	0.312453
$120$	0	0
$121$	4.83472e9	2.05039
$122$	1.54375e9	0.630896
$123$	−3.81759e8	−0.150389
$124$	7.31605e9	2.77894
$125$	0	0
$126$	−1.12683e9	−0.398281
$127$	3.11107e9	1.06119	0.530595	−	0.847625i	$-0.321968\pi$
$127$	3.11107e9	1.06119	0.530595	+	0.847625i	$0.321968\pi$
$128$	−1.45934e10	−4.80520
$129$	−1.34094e9	−0.426339
$130$	0	0
$131$	3.50469e9	1.03975	0.519874	−	0.854243i	$-0.325979\pi$
$131$	3.50469e9	1.03975	0.519874	+	0.854243i	$0.325979\pi$
$132$	9.94821e9	2.85208
$133$	3.10352e9	0.860047
$134$	−6.71878e8	−0.180019
$135$	0	0
$136$	−7.30293e9	−1.83051
$137$	−2.49029e8	−0.0603959	−0.0301980	−	0.999544i	$-0.509614\pi$
$137$	−2.49029e8	−0.0603959	−0.0301980	+	0.999544i	$0.509614\pi$
$138$	5.94311e9	1.39495
$139$	−6.24924e9	−1.41991	−0.709955	−	0.704247i	$-0.751285\pi$
$139$	−6.24924e9	−1.41991	−0.709955	+	0.704247i	$0.751285\pi$
$140$	0	0
$141$	1.45686e9	0.310408
$142$	1.76362e9	0.364006
$143$	4.97516e9	0.994936
$144$	7.17480e9	1.39052
$145$	0	0
$146$	3.11200e9	0.566829
$147$	−2.04974e9	−0.362052
$148$	6.78827e9	1.16301
$149$	5.99607e9	0.996617	0.498308	−	0.867000i	$-0.333955\pi$
$149$	5.99607e9	0.996617	0.498308	+	0.867000i	$0.333955\pi$
$150$	0	0
$151$	−1.12298e8	−0.0175783	−0.00878915	−	0.999961i	$-0.502798\pi$
$151$	−1.12298e8	−0.0175783	−0.00878915	+	0.999961i	$0.502798\pi$
$152$	−3.31593e10	−5.03859
$153$	1.15604e9	0.170554
$154$	−1.45657e10	−2.08684
$155$	0	0
$156$	6.88117e9	0.930254
$157$	−8.49455e9	−1.11581	−0.557907	−	0.829903i	$-0.688396\pi$
$157$	−8.49455e9	−1.11581	−0.557907	+	0.829903i	$0.688396\pi$
$158$	−5.17033e9	−0.660026
$159$	1.22089e9	0.151491
$160$	0	0
$161$	−6.42874e9	−0.754066
$162$	−1.90583e9	−0.217404
$163$	−7.45882e9	−0.827610	−0.413805	−	0.910366i	$-0.635800\pi$
$163$	−7.45882e9	−0.827610	−0.413805	+	0.910366i	$0.635800\pi$
$164$	−6.82526e9	−0.736753
$165$	0	0
$166$	−3.40738e10	−3.48285
$167$	1.28149e10	1.27495	0.637474	−	0.770472i	$-0.279979\pi$
$167$	1.28149e10	1.27495	0.637474	+	0.770472i	$0.279979\pi$
$168$	−1.30233e10	−1.26133
$169$	−7.16318e9	−0.675485
$170$	0	0
$171$	5.24907e9	0.469461
$172$	−2.39739e10	−2.08863
$173$	1.75478e8	0.0148941	0.00744706	−	0.999972i	$-0.497630\pi$
$173$	1.75478e8	0.0148941	0.00744706	+	0.999972i	$0.497630\pi$
$174$	1.68576e10	1.39420
$175$	0	0
$176$	9.27438e10	7.28581
$177$	9.34567e8	0.0715700
$178$	4.23733e10	3.16375
$179$	−1.98305e10	−1.44376	−0.721879	−	0.692019i	$-0.756721\pi$
$179$	−1.98305e10	−1.44376	−0.721879	+	0.692019i	$0.756721\pi$
$180$	0	0
$181$	1.37978e10	0.955558	0.477779	−	0.878480i	$-0.341442\pi$
$181$	1.37978e10	0.955558	0.477779	+	0.878480i	$0.341442\pi$
$182$	−1.00751e10	−0.680657
$183$	−2.82434e9	−0.186160
$184$	6.86873e10	4.41770
$185$	0	0
$186$	−1.81172e10	−1.10990
$187$	1.49434e10	0.893639
$188$	2.60465e10	1.52068
$189$	2.06156e9	0.117522
$190$	0	0
$191$	2.10102e10	1.14230	0.571149	−	0.820846i	$-0.306498\pi$
$191$	2.10102e10	1.14230	0.571149	+	0.820846i	$0.306498\pi$
$192$	5.21729e10	2.77070
$193$	−3.23747e10	−1.67957	−0.839784	−	0.542920i	$-0.817319\pi$
$193$	−3.23747e10	−1.67957	−0.839784	+	0.542920i	$0.817319\pi$
$194$	−2.18454e10	−1.10727
$195$	0	0
$196$	−3.66462e10	−1.77368
$197$	4.18256e9	0.197854	0.0989268	−	0.995095i	$-0.468459\pi$
$197$	4.18256e9	0.197854	0.0989268	+	0.995095i	$0.468459\pi$
$198$	−2.46354e10	−1.13911
$199$	2.68922e10	1.21559	0.607795	−	0.794094i	$-0.292054\pi$
$199$	2.68922e10	1.21559	0.607795	+	0.794094i	$0.292054\pi$
$200$	0	0
$201$	1.22922e9	0.0531188
$202$	−3.53763e10	−1.49496
$203$	−1.82351e10	−0.753659
$204$	2.06683e10	0.835543
$205$	0	0
$206$	−9.23416e10	−3.57269
$207$	−1.08731e10	−0.411611
$208$	6.41508e10	2.37639
$209$	6.78512e10	2.45980
$210$	0	0
$211$	−3.91894e10	−1.36112	−0.680561	−	0.732691i	$-0.738264\pi$
$211$	−3.91894e10	−1.36112	−0.680561	+	0.732691i	$0.738264\pi$
$212$	2.18275e10	0.742153
$213$	−3.22660e9	−0.107408
$214$	−5.15446e10	−1.68005
$215$	0	0
$216$	−2.20266e10	−0.688503
$217$	1.95976e10	0.599977
$218$	1.95305e10	0.585678
$219$	−5.69351e9	−0.167256
$220$	0	0
$221$	1.03363e10	0.291475
$222$	−1.68103e10	−0.464500
$223$	−3.16572e10	−0.857236	−0.428618	−	0.903486i	$-0.640999\pi$
$223$	−3.16572e10	−0.857236	−0.428618	+	0.903486i	$0.640999\pi$
$224$	−1.05494e11	−2.79969
$225$	0	0
$226$	1.29698e10	0.330708
$227$	−2.25669e10	−0.564100	−0.282050	−	0.959400i	$-0.591014\pi$
$227$	−2.25669e10	−0.564100	−0.282050	+	0.959400i	$0.591014\pi$
$228$	9.38453e10	2.29988
$229$	−4.23471e10	−1.01757	−0.508785	−	0.860894i	$-0.669905\pi$
$229$	−4.23471e10	−1.01757	−0.508785	+	0.860894i	$0.669905\pi$
$230$	0	0
$231$	2.66485e10	0.615769
$232$	1.94831e11	4.41532
$233$	−1.85845e10	−0.413094	−0.206547	−	0.978437i	$-0.566223\pi$
$233$	−1.85845e10	−0.413094	−0.206547	+	0.978437i	$0.566223\pi$
$234$	−1.70403e10	−0.371541
$235$	0	0
$236$	1.67086e10	0.350620
$237$	9.45928e9	0.194756
$238$	−3.02616e10	−0.611358
$239$	9.95998e10	1.97455	0.987275	−	0.159025i	$-0.0508350\pi$
$239$	9.95998e10	1.97455	0.987275	+	0.159025i	$0.0508350\pi$
$240$	0	0
$241$	−3.44580e9	−0.0657982	−0.0328991	−	0.999459i	$-0.510474\pi$
$241$	−3.44580e9	−0.0657982	−0.0328991	+	0.999459i	$0.510474\pi$
$242$	−2.14051e11	−4.01187
$243$	3.48678e9	0.0641500
$244$	−5.04948e10	−0.911995
$245$	0	0
$246$	1.69018e10	0.294257
$247$	4.69326e10	0.802303
$248$	−2.09389e11	−3.51497
$249$	6.23391e10	1.02769
$250$	0	0
$251$	1.23135e9	0.0195816	0.00979082	−	0.999952i	$-0.496883\pi$
$251$	1.23135e9	0.0195816	0.00979082	+	0.999952i	$0.496883\pi$
$252$	3.68576e10	0.575738
$253$	−1.40549e11	−2.15668
$254$	−1.37738e11	−2.07636
$255$	0	0
$256$	3.16318e11	4.60303
$257$	2.31623e10	0.331193	0.165597	−	0.986194i	$-0.447045\pi$
$257$	2.31623e10	0.331193	0.165597	+	0.986194i	$0.447045\pi$
$258$	5.93683e10	0.834191
$259$	1.81839e10	0.251095
$260$	0	0
$261$	−3.08415e10	−0.411389
$262$	−1.55165e11	−2.03441
$263$	−9.41772e10	−1.21379	−0.606897	−	0.794781i	$-0.707586\pi$
$263$	−9.41772e10	−1.21379	−0.606897	+	0.794781i	$0.707586\pi$
$264$	−2.84723e11	−3.60749
$265$	0	0
$266$	−1.37404e11	−1.68280
$267$	−7.75232e10	−0.933536
$268$	2.19766e10	0.260228
$269$	1.96650e10	0.228986	0.114493	−	0.993424i	$-0.463476\pi$
$269$	1.96650e10	0.228986	0.114493	+	0.993424i	$0.463476\pi$
$270$	0	0
$271$	1.53707e11	1.73114	0.865568	−	0.500792i	$-0.166958\pi$
$271$	1.53707e11	1.73114	0.865568	+	0.500792i	$0.166958\pi$
$272$	1.92683e11	2.13444
$273$	1.84327e10	0.200843
$274$	1.10254e10	0.118173
$275$	0	0
$276$	−1.94394e11	−2.01648
$277$	−9.15920e9	−0.0934757	−0.0467378	−	0.998907i	$-0.514883\pi$
$277$	−9.15920e9	−0.0934757	−0.0467378	+	0.998907i	$0.514883\pi$
$278$	2.76677e11	2.77825
$279$	3.31460e10	0.327501
$280$	0	0
$281$	−9.39374e10	−0.898794	−0.449397	−	0.893332i	$-0.648361\pi$
$281$	−9.39374e10	−0.898794	−0.449397	+	0.893332i	$0.648361\pi$
$282$	−6.45007e10	−0.607356
$283$	1.62185e11	1.50304	0.751520	−	0.659710i	$-0.229321\pi$
$283$	1.62185e11	1.50304	0.751520	+	0.659710i	$0.229321\pi$
$284$	−5.76867e10	−0.526190
$285$	0	0
$286$	−2.20269e11	−1.94673
$287$	−1.82829e10	−0.159066
$288$	−1.78425e11	−1.52823
$289$	−8.75416e10	−0.738201
$290$	0	0
$291$	3.99669e10	0.326724
$292$	−1.01791e11	−0.819383
$293$	3.62588e10	0.287415	0.143707	−	0.989620i	$-0.454098\pi$
$293$	3.62588e10	0.287415	0.143707	+	0.989620i	$0.454098\pi$
$294$	9.07494e10	0.708404
$295$	0	0
$296$	−1.94284e11	−1.47104
$297$	4.50713e10	0.336121
$298$	−2.65468e11	−1.95002
$299$	−9.72179e10	−0.703438
$300$	0	0
$301$	−6.42194e10	−0.450938
$302$	4.97185e9	0.0343944
$303$	6.47220e10	0.441123
$304$	8.74888e11	5.87518
$305$	0	0
$306$	−5.11823e10	−0.333713
$307$	2.54744e11	1.63675	0.818374	−	0.574686i	$-0.194876\pi$
$307$	2.54744e11	1.63675	0.818374	+	0.574686i	$0.194876\pi$
$308$	4.76433e11	3.01664
$309$	1.68942e11	1.05420
$310$	0	0
$311$	2.48364e11	1.50545	0.752725	−	0.658335i	$-0.228739\pi$
$311$	2.48364e11	1.50545	0.752725	+	0.658335i	$0.228739\pi$
$312$	−1.96943e11	−1.17664
$313$	−2.07235e11	−1.22043	−0.610216	−	0.792235i	$-0.708917\pi$
$313$	−2.07235e11	−1.22043	−0.610216	+	0.792235i	$0.708917\pi$
$314$	3.76085e11	2.18324
$315$	0	0
$316$	1.69117e11	0.954105
$317$	−4.95903e10	−0.275823	−0.137911	−	0.990445i	$-0.544039\pi$
$317$	−4.95903e10	−0.275823	−0.137911	+	0.990445i	$0.544039\pi$
$318$	−5.40530e10	−0.296413
$319$	−3.98667e11	−2.15552
$320$	0	0
$321$	9.43025e10	0.495736
$322$	2.84624e11	1.47543
$323$	1.40967e11	0.720619
$324$	6.23383e10	0.314270
$325$	0	0
$326$	3.30229e11	1.61933
$327$	−3.57316e10	−0.172818
$328$	1.95343e11	0.931890
$329$	6.97712e10	0.328318
$330$	0	0
$331$	−2.76790e11	−1.26743	−0.633715	−	0.773567i	$-0.718471\pi$
$331$	−2.76790e11	−1.26743	−0.633715	+	0.773567i	$0.718471\pi$
$332$	1.11453e12	5.03465
$333$	3.07549e10	0.137061
$334$	−5.67364e11	−2.49461
$335$	0	0
$336$	3.43611e11	1.47075
$337$	1.04619e11	0.441850	0.220925	−	0.975291i	$-0.429092\pi$
$337$	1.04619e11	0.441850	0.220925	+	0.975291i	$0.429092\pi$
$338$	3.17140e11	1.32168
$339$	−2.37287e10	−0.0975831
$340$	0	0
$341$	4.28456e11	1.71598
$342$	−2.32395e11	−0.918566
$343$	−2.54704e11	−0.993603
$344$	6.86147e11	2.64182
$345$	0	0
$346$	−7.76905e9	−0.0291424
$347$	−2.19698e11	−0.813475	−0.406738	−	0.913545i	$-0.633334\pi$
$347$	−2.19698e11	−0.813475	−0.406738	+	0.913545i	$0.633334\pi$
$348$	−5.51398e11	−2.01539
$349$	−5.84554e10	−0.210916	−0.105458	−	0.994424i	$-0.533631\pi$
$349$	−5.84554e10	−0.210916	−0.105458	+	0.994424i	$0.533631\pi$
$350$	0	0
$351$	3.11758e10	0.109632
$352$	−2.30637e12	−8.00733
$353$	2.01385e11	0.690305	0.345153	−	0.938547i	$-0.387827\pi$
$353$	2.01385e11	0.690305	0.345153	+	0.938547i	$0.387827\pi$
$354$	−4.13767e10	−0.140036
$355$	0	0
$356$	−1.38599e12	−4.57337
$357$	5.53645e10	0.180395
$358$	8.77967e11	2.82491
$359$	−6.91613e10	−0.219755	−0.109877	−	0.993945i	$-0.535046\pi$
$359$	−6.91613e10	−0.219755	−0.109877	+	0.993945i	$0.535046\pi$
$360$	0	0
$361$	3.17378e11	0.983547
$362$	−6.10880e11	−1.86968
$363$	3.91612e11	1.18380
$364$	3.29549e11	0.983928
$365$	0	0
$366$	1.25044e11	0.364248
$367$	5.71203e11	1.64359	0.821794	−	0.569784i	$-0.192973\pi$
$367$	5.71203e11	1.64359	0.821794	+	0.569784i	$0.192973\pi$
$368$	−1.81227e12	−5.15120
$369$	−3.09225e10	−0.0868271
$370$	0	0
$371$	5.84698e10	0.160232
$372$	5.92600e11	1.60442
$373$	3.02470e11	0.809083	0.404541	−	0.914520i	$-0.367431\pi$
$373$	3.02470e11	0.809083	0.404541	+	0.914520i	$0.367431\pi$
$374$	−6.61599e11	−1.74853
$375$	0	0
$376$	−7.45465e11	−1.92345
$377$	−2.75758e11	−0.703059
$378$	−9.12730e10	−0.229948
$379$	−5.51774e11	−1.37368	−0.686839	−	0.726810i	$-0.741002\pi$
$379$	−5.51774e11	−1.37368	−0.686839	+	0.726810i	$0.741002\pi$
$380$	0	0
$381$	2.51997e11	0.612679
$382$	−9.30197e11	−2.23506
$383$	1.72781e9	0.00410299	0.00205150	−	0.999998i	$-0.499347\pi$
$383$	1.72781e9	0.00410299	0.00205150	+	0.999998i	$0.499347\pi$
$384$	−1.18206e12	−2.77428
$385$	0	0
$386$	1.43335e12	3.28631
$387$	−1.08616e11	−0.246147
$388$	7.14546e11	1.60062
$389$	2.10497e11	0.466094	0.233047	−	0.972466i	$-0.425130\pi$
$389$	2.10497e11	0.466094	0.233047	+	0.972466i	$0.425130\pi$
$390$	0	0
$391$	−2.92004e11	−0.631819
$392$	1.04883e12	2.24346
$393$	2.83880e11	0.600299
$394$	−1.85177e11	−0.387127
$395$	0	0
$396$	8.05805e11	1.64665
$397$	−2.43920e11	−0.492823	−0.246411	−	0.969165i	$-0.579251\pi$
$397$	−2.43920e11	−0.492823	−0.246411	+	0.969165i	$0.579251\pi$
$398$	−1.19061e12	−2.37847
$399$	2.51385e11	0.496548
$400$	0	0
$401$	−8.42048e11	−1.62625	−0.813125	−	0.582089i	$-0.802235\pi$
$401$	−8.42048e11	−1.62625	−0.813125	+	0.582089i	$0.802235\pi$
$402$	−5.44221e10	−0.103934
$403$	2.96363e11	0.559695
$404$	1.15713e12	2.16105
$405$	0	0
$406$	8.07333e11	1.47464
$407$	3.97548e11	0.718149
$408$	−5.91538e11	−1.05685
$409$	−6.25516e11	−1.10531	−0.552655	−	0.833410i	$-0.686385\pi$
$409$	−6.25516e11	−1.10531	−0.552655	+	0.833410i	$0.686385\pi$
$410$	0	0
$411$	−2.01714e10	−0.0348696
$412$	3.02042e12	5.16452
$413$	4.47577e10	0.0756994
$414$	4.81392e11	0.805374
$415$	0	0
$416$	−1.59532e12	−2.61172
$417$	−5.06189e11	−0.819786
$418$	−3.00402e12	−4.81293
$419$	4.41203e11	0.699318	0.349659	−	0.936877i	$-0.386297\pi$
$419$	4.41203e11	0.699318	0.349659	+	0.936877i	$0.386297\pi$
$420$	0	0
$421$	−7.82467e11	−1.21394	−0.606969	−	0.794726i	$-0.707615\pi$
$421$	−7.82467e11	−1.21394	−0.606969	+	0.794726i	$0.707615\pi$
$422$	1.73506e12	2.66322
$423$	1.18006e11	0.179214
$424$	−6.24716e11	−0.938720
$425$	0	0
$426$	1.42853e11	0.210159
$427$	−1.35261e11	−0.196901
$428$	1.68598e12	2.42860
$429$	4.02988e11	0.574427
$430$	0	0
$431$	2.81183e11	0.392501	0.196251	−	0.980554i	$-0.437123\pi$
$431$	2.81183e11	0.392501	0.196251	+	0.980554i	$0.437123\pi$
$432$	5.81159e11	0.802820
$433$	3.67906e11	0.502969	0.251485	−	0.967861i	$-0.419081\pi$
$433$	3.67906e11	0.502969	0.251485	+	0.967861i	$0.419081\pi$
$434$	−8.67658e11	−1.17394
$435$	0	0
$436$	−6.38826e11	−0.846629
$437$	−1.32586e12	−1.73912
$438$	2.52072e11	0.327259
$439$	−5.56286e11	−0.714838	−0.357419	−	0.933944i	$-0.616343\pi$
$439$	−5.56286e11	−0.714838	−0.357419	+	0.933944i	$0.616343\pi$
$440$	0	0
$441$	−1.66029e11	−0.209031
$442$	−4.57628e11	−0.570311
$443$	9.78779e11	1.20745	0.603723	−	0.797194i	$-0.293683\pi$
$443$	9.78779e11	1.20745	0.603723	+	0.797194i	$0.293683\pi$
$444$	5.49850e11	0.671461
$445$	0	0
$446$	1.40158e12	1.67730
$447$	4.85681e11	0.575397
$448$	2.49863e12	2.93056
$449$	4.25426e11	0.493988	0.246994	−	0.969017i	$-0.420557\pi$
$449$	4.25426e11	0.493988	0.246994	+	0.969017i	$0.420557\pi$
$450$	0	0
$451$	−3.99714e11	−0.454940
$452$	−4.24232e11	−0.478057
$453$	−9.09616e9	−0.0101488
$454$	9.99120e11	1.10374
$455$	0	0
$456$	−2.68590e12	−2.90903
$457$	1.22187e12	1.31040	0.655198	−	0.755457i	$-0.272585\pi$
$457$	1.22187e12	1.31040	0.655198	+	0.755457i	$0.272585\pi$
$458$	1.87486e12	1.99102
$459$	9.36396e10	0.0984697
$460$	0	0
$461$	−9.93468e11	−1.02447	−0.512236	−	0.858845i	$-0.671183\pi$
$461$	−9.93468e11	−1.02447	−0.512236	+	0.858845i	$0.671183\pi$
$462$	−1.17982e12	−1.20484
$463$	−1.19400e12	−1.20751	−0.603756	−	0.797169i	$-0.706330\pi$
$463$	−1.19400e12	−1.20751	−0.603756	+	0.797169i	$0.706330\pi$
$464$	−5.14050e12	−5.14842
$465$	0	0
$466$	8.22802e11	0.808274
$467$	−5.96205e11	−0.580056	−0.290028	−	0.957018i	$-0.593665\pi$
$467$	−5.96205e11	−0.580056	−0.290028	+	0.957018i	$0.593665\pi$
$468$	5.57375e11	0.537083
$469$	5.88691e10	0.0561836
$470$	0	0
$471$	−6.88059e11	−0.644216
$472$	−4.78210e11	−0.443485
$473$	−1.40401e12	−1.28972
$474$	−4.18797e11	−0.381066
$475$	0	0
$476$	9.89832e11	0.883752
$477$	9.88917e10	0.0874636
$478$	−4.40965e12	−3.86348
$479$	2.04094e12	1.77142	0.885709	−	0.464240i	$-0.153673\pi$
$479$	2.04094e12	1.77142	0.885709	+	0.464240i	$0.153673\pi$
$480$	0	0
$481$	2.74984e11	0.234236
$482$	1.52558e11	0.128743
$483$	−5.20728e11	−0.435360
$484$	7.00142e12	5.79939
$485$	0	0
$486$	−1.54373e11	−0.125518
$487$	−1.96613e12	−1.58391	−0.791957	−	0.610577i	$-0.790938\pi$
$487$	−1.96613e12	−1.58391	−0.791957	+	0.610577i	$0.790938\pi$
$488$	1.44519e12	1.15355
$489$	−6.04164e11	−0.477821
$490$	0	0
$491$	−1.24449e12	−0.966332	−0.483166	−	0.875529i	$-0.660513\pi$
$491$	−1.24449e12	−0.966332	−0.483166	+	0.875529i	$0.660513\pi$
$492$	−5.52846e11	−0.425364
$493$	−8.28266e11	−0.631479
$494$	−2.07788e12	−1.56982
$495$	0	0
$496$	5.52461e12	4.09858
$497$	−1.54526e11	−0.113605
$498$	−2.75998e12	−2.01082
$499$	−6.90063e11	−0.498237	−0.249119	−	0.968473i	$-0.580141\pi$
$499$	−6.90063e11	−0.498237	−0.249119	+	0.968473i	$0.580141\pi$
$500$	0	0
$501$	1.03801e12	0.736092
$502$	−5.45162e10	−0.0383141
$503$	−1.58581e12	−1.10458	−0.552289	−	0.833653i	$-0.686245\pi$
$503$	−1.58581e12	−1.10458	−0.552289	+	0.833653i	$0.686245\pi$
$504$	−1.05488e12	−0.728228
$505$	0	0
$506$	6.22263e12	4.21985
$507$	−5.80218e11	−0.389991
$508$	4.50531e12	3.00150
$509$	−4.16173e11	−0.274817	−0.137409	−	0.990514i	$-0.543877\pi$
$509$	−4.16173e11	−0.274817	−0.137409	+	0.990514i	$0.543877\pi$
$510$	0	0
$511$	−2.72670e11	−0.176906
$512$	−6.53274e12	−4.20127
$513$	4.25175e11	0.271044
$514$	−1.02548e12	−0.648025
$515$	0	0
$516$	−1.94189e12	−1.20587
$517$	1.52538e12	0.939013
$518$	−8.05066e11	−0.491301
$519$	1.42137e10	0.00859913
$520$	0	0
$521$	−1.49363e12	−0.888121	−0.444060	−	0.895997i	$-0.646462\pi$
$521$	−1.49363e12	−0.888121	−0.444060	+	0.895997i	$0.646462\pi$
$522$	1.36546e12	0.804940
$523$	3.43582e11	0.200804	0.100402	−	0.994947i	$-0.467987\pi$
$523$	3.43582e11	0.200804	0.100402	+	0.994947i	$0.467987\pi$
$524$	5.07533e12	2.94085
$525$	0	0
$526$	4.16957e12	2.37495
$527$	8.90156e11	0.502711
$528$	7.51225e12	4.20646
$529$	9.45273e11	0.524815
$530$	0	0
$531$	7.56999e10	0.0413209
$532$	4.49438e12	2.43258
$533$	−2.76482e11	−0.148386
$534$	3.43223e12	1.82659
$535$	0	0
$536$	−6.28982e11	−0.329152
$537$	−1.60627e12	−0.833554
$538$	−8.70642e11	−0.448042
$539$	−2.14614e12	−1.09524
$540$	0	0
$541$	−2.79621e12	−1.40340	−0.701702	−	0.712471i	$-0.747576\pi$
$541$	−2.79621e12	−1.40340	−0.701702	+	0.712471i	$0.747576\pi$
$542$	−6.80516e12	−3.38720
$543$	1.11762e12	0.551692
$544$	−4.79169e12	−2.34582
$545$	0	0
$546$	−8.16083e11	−0.392978
$547$	−1.58416e12	−0.756580	−0.378290	−	0.925687i	$-0.623488\pi$
$547$	−1.58416e12	−0.756580	−0.378290	+	0.925687i	$0.623488\pi$
$548$	−3.60633e11	−0.170825
$549$	−2.28771e11	−0.107480
$550$	0	0
$551$	−3.76078e12	−1.73818
$552$	5.56367e12	2.55056
$553$	4.53017e11	0.205993
$554$	4.05511e11	0.182898
$555$	0	0
$556$	−9.04987e12	−4.01611
$557$	−1.94007e12	−0.854023	−0.427011	−	0.904246i	$-0.640434\pi$
$557$	−1.94007e12	−0.854023	−0.427011	+	0.904246i	$0.640434\pi$
$558$	−1.46750e12	−0.640800
$559$	−9.71151e11	−0.420662
$560$	0	0
$561$	1.21042e12	0.515943
$562$	4.15895e12	1.75861
$563$	−4.26047e12	−1.78719	−0.893593	−	0.448879i	$-0.851823\pi$
$563$	−4.26047e12	−1.78719	−0.893593	+	0.448879i	$0.851823\pi$
$564$	2.10977e12	0.877967
$565$	0	0
$566$	−7.18050e12	−2.94090
$567$	1.66987e11	0.0678513
$568$	1.65102e12	0.665558
$569$	2.53770e12	1.01493	0.507464	−	0.861673i	$-0.330583\pi$
$569$	2.53770e12	1.01493	0.507464	+	0.861673i	$0.330583\pi$
$570$	0	0
$571$	2.23395e12	0.879449	0.439725	−	0.898133i	$-0.355076\pi$
$571$	2.23395e12	0.879449	0.439725	+	0.898133i	$0.355076\pi$
$572$	7.20481e12	2.81410
$573$	1.70182e12	0.659506
$574$	8.09452e11	0.311234
$575$	0	0
$576$	4.22600e12	1.59966
$577$	−2.89653e12	−1.08789	−0.543947	−	0.839119i	$-0.683071\pi$
$577$	−2.89653e12	−1.08789	−0.543947	+	0.839119i	$0.683071\pi$
$578$	3.87579e12	1.44439
$579$	−2.62235e12	−0.969699
$580$	0	0
$581$	2.98550e12	1.08699
$582$	−1.76948e12	−0.639281
$583$	1.27831e12	0.458275
$584$	2.91332e12	1.03641
$585$	0	0
$586$	−1.60531e12	−0.562366
$587$	3.00835e12	1.04582	0.522910	−	0.852388i	$-0.324846\pi$
$587$	3.00835e12	1.04582	0.522910	+	0.852388i	$0.324846\pi$
$588$	−2.96834e12	−1.02404
$589$	4.04179e12	1.38374
$590$	0	0
$591$	3.38787e11	0.114231
$592$	5.12607e12	1.71529
$593$	4.23442e12	1.40620	0.703102	−	0.711089i	$-0.251798\pi$
$593$	4.23442e12	1.40620	0.703102	+	0.711089i	$0.251798\pi$
$594$	−1.99547e12	−0.657667
$595$	0	0
$596$	8.68323e12	2.81886
$597$	2.17827e12	0.701822
$598$	4.30419e12	1.37637
$599$	−9.00803e11	−0.285897	−0.142948	−	0.989730i	$-0.545658\pi$
$599$	−9.00803e11	−0.285897	−0.142948	+	0.989730i	$0.545658\pi$
$600$	0	0
$601$	5.71501e12	1.78683	0.893413	−	0.449237i	$-0.148304\pi$
$601$	5.71501e12	1.78683	0.893413	+	0.449237i	$0.148304\pi$
$602$	2.84323e12	0.882322
$603$	9.95669e10	0.0306681
$604$	−1.62625e11	−0.0497190
$605$	0	0
$606$	−2.86548e12	−0.863118
$607$	−3.28972e11	−0.0983580	−0.0491790	−	0.998790i	$-0.515660\pi$
$607$	−3.28972e11	−0.0983580	−0.0491790	+	0.998790i	$0.515660\pi$
$608$	−2.17569e13	−6.45700
$609$	−1.47704e12	−0.435125
$610$	0	0
$611$	1.05511e12	0.306275
$612$	1.67413e12	0.482401
$613$	6.62872e12	1.89608	0.948042	−	0.318146i	$-0.103060\pi$
$613$	6.62872e12	1.89608	0.948042	+	0.318146i	$0.103060\pi$
$614$	−1.12785e13	−3.20252
$615$	0	0
$616$	−1.36358e13	−3.81563
$617$	1.88213e12	0.522838	0.261419	−	0.965225i	$-0.415810\pi$
$617$	1.88213e12	0.522838	0.261419	+	0.965225i	$0.415810\pi$
$618$	−7.47967e12	−2.06269
$619$	−1.47161e12	−0.402889	−0.201444	−	0.979500i	$-0.564564\pi$
$619$	−1.47161e12	−0.402889	−0.201444	+	0.979500i	$0.564564\pi$
$620$	0	0
$621$	−8.80722e11	−0.237644
$622$	−1.09960e13	−2.94562
$623$	−3.71269e12	−0.987398
$624$	5.19622e12	1.37201
$625$	0	0
$626$	9.17504e12	2.38794
$627$	5.49595e12	1.42016
$628$	−1.23014e13	−3.15600
$629$	8.25941e11	0.210388
$630$	0	0
$631$	−1.80892e12	−0.454241	−0.227121	−	0.973867i	$-0.572931\pi$
$631$	−1.80892e12	−0.454241	−0.227121	+	0.973867i	$0.572931\pi$
$632$	−4.84023e12	−1.20681
$633$	−3.17434e12	−0.785844
$634$	2.19554e12	0.539685
$635$	0	0
$636$	1.76803e12	0.428482
$637$	−1.48449e12	−0.357231
$638$	1.76504e13	4.21757
$639$	−2.61355e11	−0.0620121
$640$	0	0
$641$	1.20331e12	0.281525	0.140762	−	0.990043i	$-0.455045\pi$
$641$	1.20331e12	0.281525	0.140762	+	0.990043i	$0.455045\pi$
$642$	−4.17512e12	−0.969976
$643$	−4.90684e11	−0.113202	−0.0566008	−	0.998397i	$-0.518026\pi$
$643$	−4.90684e11	−0.113202	−0.0566008	+	0.998397i	$0.518026\pi$
$644$	−9.30981e12	−2.13282
$645$	0	0
$646$	−6.24111e12	−1.40999
$647$	−3.98948e12	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$647$	−3.98948e12	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$648$	−1.78416e12	−0.397508
$649$	9.78522e11	0.216506
$650$	0	0
$651$	1.58741e12	0.346397
$652$	−1.08015e13	−2.34084
$653$	1.11005e12	0.238909	0.119454	−	0.992840i	$-0.461886\pi$
$653$	1.11005e12	0.238909	0.119454	+	0.992840i	$0.461886\pi$
$654$	1.58197e12	0.338141
$655$	0	0
$656$	−5.15399e12	−1.08662
$657$	−4.61174e11	−0.0965653
$658$	−3.08902e12	−0.642399
$659$	−1.17596e12	−0.242890	−0.121445	−	0.992598i	$-0.538753\pi$
$659$	−1.17596e12	−0.242890	−0.121445	+	0.992598i	$0.538753\pi$
$660$	0	0
$661$	6.46397e12	1.31702	0.658511	−	0.752571i	$-0.271187\pi$
$661$	6.46397e12	1.31702	0.658511	+	0.752571i	$0.271187\pi$
$662$	1.22545e13	2.47990
$663$	8.37244e11	0.168283
$664$	−3.18984e13	−6.36814
$665$	0	0
$666$	−1.36163e12	−0.268179
$667$	7.79021e12	1.52399
$668$	1.85580e13	3.60610
$669$	−2.56423e12	−0.494925
$670$	0	0
$671$	−2.95717e12	−0.563152
$672$	−8.54499e12	−1.61640
$673$	9.92816e12	1.86552	0.932762	−	0.360492i	$-0.117391\pi$
$673$	9.92816e12	1.86552	0.932762	+	0.360492i	$0.117391\pi$
$674$	−4.63185e12	−0.864540
$675$	0	0
$676$	−1.03734e13	−1.91056
$677$	4.10285e12	0.750649	0.375324	−	0.926894i	$-0.377531\pi$
$677$	4.10285e12	0.750649	0.375324	+	0.926894i	$0.377531\pi$
$678$	1.05055e12	0.190935
$679$	1.91407e12	0.345576
$680$	0	0
$681$	−1.82792e12	−0.325683
$682$	−1.89693e13	−3.35755
$683$	−6.13606e12	−1.07894	−0.539469	−	0.842005i	$-0.681375\pi$
$683$	−6.13606e12	−1.07894	−0.539469	+	0.842005i	$0.681375\pi$
$684$	7.60147e12	1.32784
$685$	0	0
$686$	1.12767e13	1.94412
$687$	−3.43012e12	−0.587495
$688$	−1.81036e13	−3.08046
$689$	8.84204e11	0.149474
$690$	0	0
$691$	1.15337e13	1.92449	0.962245	−	0.272184i	$-0.0877459\pi$
$691$	1.15337e13	1.92449	0.962245	+	0.272184i	$0.0877459\pi$
$692$	2.54119e11	0.0421270
$693$	2.15852e12	0.355515
$694$	9.72685e12	1.59168
$695$	0	0
$696$	1.57813e13	2.54919
$697$	−8.30440e11	−0.133279
$698$	2.58803e12	0.412687
$699$	−1.50534e12	−0.238500
$700$	0	0
$701$	4.38615e12	0.686044	0.343022	−	0.939327i	$-0.388549\pi$
$701$	4.38615e12	0.686044	0.343022	+	0.939327i	$0.388549\pi$
$702$	−1.38027e12	−0.214509
$703$	3.75022e12	0.579106
$704$	5.46267e13	8.38162
$705$	0	0
$706$	−8.91605e12	−1.35068
$707$	3.09962e12	0.466575
$708$	1.35340e12	0.202430
$709$	3.60073e12	0.535158	0.267579	−	0.963536i	$-0.413776\pi$
$709$	3.60073e12	0.535158	0.267579	+	0.963536i	$0.413776\pi$
$710$	0	0
$711$	7.66201e11	0.112442
$712$	3.96679e13	5.78468
$713$	−8.37231e12	−1.21323
$714$	−2.45119e12	−0.352968
$715$	0	0
$716$	−2.87176e13	−4.08356
$717$	8.06758e12	1.14001
$718$	3.06202e12	0.429980
$719$	−5.19426e12	−0.724842	−0.362421	−	0.932014i	$-0.618050\pi$
$719$	−5.19426e12	−0.724842	−0.362421	+	0.932014i	$0.618050\pi$
$720$	0	0
$721$	8.09085e12	1.11503
$722$	−1.40515e13	−1.92444
$723$	−2.79110e11	−0.0379886
$724$	1.99814e13	2.70273
$725$	0	0
$726$	−1.73381e13	−2.31626
$727$	8.41803e12	1.11765	0.558825	−	0.829286i	$-0.311252\pi$
$727$	8.41803e12	1.11765	0.558825	+	0.829286i	$0.311252\pi$
$728$	−9.43186e12	−1.24453
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	−2.91695e12	−0.377833
$732$	−4.09008e12	−0.526541
$733$	3.22252e12	0.412314	0.206157	−	0.978519i	$-0.433904\pi$
$733$	3.22252e12	0.412314	0.206157	+	0.978519i	$0.433904\pi$
$734$	−2.52892e13	−3.21591
$735$	0	0
$736$	4.50680e13	5.66133
$737$	1.28703e12	0.160689
$738$	1.36905e12	0.169889
$739$	9.19731e12	1.13439	0.567193	−	0.823585i	$-0.308029\pi$
$739$	9.19731e12	1.13439	0.567193	+	0.823585i	$0.308029\pi$
$740$	0	0
$741$	3.80154e12	0.463210
$742$	−2.58867e12	−0.313516
$743$	−4.23199e12	−0.509443	−0.254721	−	0.967014i	$-0.581984\pi$
$743$	−4.23199e12	−0.509443	−0.254721	+	0.967014i	$0.581984\pi$
$744$	−1.69605e13	−2.02937
$745$	0	0
$746$	−1.33915e13	−1.58308
$747$	5.04947e12	0.593339
$748$	2.16404e13	2.52759
$749$	4.51627e12	0.524339
$750$	0	0
$751$	−1.15200e13	−1.32152	−0.660758	−	0.750599i	$-0.729765\pi$
$751$	−1.15200e13	−1.32152	−0.660758	+	0.750599i	$0.729765\pi$
$752$	1.96686e13	2.24282
$753$	9.97392e10	0.0113055
$754$	1.22088e13	1.37563
$755$	0	0
$756$	2.98547e12	0.332402
$757$	−1.01569e13	−1.12417	−0.562085	−	0.827080i	$-0.690000\pi$
$757$	−1.01569e13	−1.12417	−0.562085	+	0.827080i	$0.690000\pi$
$758$	2.44290e13	2.68779
$759$	−1.13845e13	−1.24516
$760$	0	0
$761$	−6.15777e12	−0.665568	−0.332784	−	0.943003i	$-0.607988\pi$
$761$	−6.15777e12	−0.665568	−0.332784	+	0.943003i	$0.607988\pi$
$762$	−1.11568e13	−1.19879
$763$	−1.71123e12	−0.182789
$764$	3.04260e13	3.23091
$765$	0	0
$766$	−7.64963e10	−0.00802806
$767$	6.76843e11	0.0706169
$768$	2.56218e13	2.65756
$769$	−1.56690e12	−0.161575	−0.0807873	−	0.996731i	$-0.525743\pi$
$769$	−1.56690e12	−0.161575	−0.0807873	+	0.996731i	$0.525743\pi$
$770$	0	0
$771$	1.87614e12	0.191215
$772$	−4.68836e13	−4.75054
$773$	−1.55751e13	−1.56900	−0.784501	−	0.620128i	$-0.787081\pi$
$773$	−1.55751e13	−1.56900	−0.784501	+	0.620128i	$0.787081\pi$
$774$	4.80883e12	0.481621
$775$	0	0
$776$	−2.04507e13	−2.02456
$777$	1.47289e12	0.144970
$778$	−9.31948e12	−0.911976
$779$	−3.77065e12	−0.366858
$780$	0	0
$781$	−3.37836e12	−0.324919
$782$	1.29281e13	1.23624
$783$	−2.49816e12	−0.237516
$784$	−2.76728e13	−2.61596
$785$	0	0
$786$	−1.25684e13	−1.17457
$787$	−1.57345e12	−0.146206	−0.0731030	−	0.997324i	$-0.523290\pi$
$787$	−1.57345e12	−0.146206	−0.0731030	+	0.997324i	$0.523290\pi$
$788$	6.05699e12	0.559614
$789$	−7.62835e12	−0.700784
$790$	0	0
$791$	−1.13640e12	−0.103213
$792$	−2.30626e13	−2.08279
$793$	−2.04547e12	−0.183681
$794$	1.07992e13	0.964275
$795$	0	0
$796$	3.89441e13	3.43821
$797$	−2.06332e13	−1.81136	−0.905678	−	0.423965i	$-0.860638\pi$
$797$	−2.06332e13	−1.81136	−0.905678	+	0.423965i	$0.860638\pi$
$798$	−1.11297e13	−0.971564
$799$	3.16912e12	0.275092
$800$	0	0
$801$	−6.27938e12	−0.538977
$802$	3.72805e13	3.18198
$803$	−5.96129e12	−0.505965
$804$	1.78010e12	0.150243
$805$	0	0
$806$	−1.31211e13	−1.09512
$807$	1.59287e12	0.132205
$808$	−3.31177e13	−2.73343
$809$	2.00299e13	1.64403	0.822015	−	0.569466i	$-0.192850\pi$
$809$	2.00299e13	1.64403	0.822015	+	0.569466i	$0.192850\pi$
$810$	0	0
$811$	1.91792e12	0.155681	0.0778406	−	0.996966i	$-0.475197\pi$
$811$	1.91792e12	0.155681	0.0778406	+	0.996966i	$0.475197\pi$
$812$	−2.64072e13	−2.13167
$813$	1.24502e13	0.999472
$814$	−1.76009e13	−1.40516
$815$	0	0
$816$	1.56074e13	1.23232
$817$	−1.32445e13	−1.04001
$818$	2.76939e13	2.16269
$819$	1.49305e12	0.115957
$820$	0	0
$821$	5.52525e12	0.424432	0.212216	−	0.977223i	$-0.431932\pi$
$821$	5.52525e12	0.424432	0.212216	+	0.977223i	$0.431932\pi$
$822$	8.93059e11	0.0682271
$823$	−2.95173e12	−0.224273	−0.112137	−	0.993693i	$-0.535769\pi$
$823$	−2.95173e12	−0.224273	−0.112137	+	0.993693i	$0.535769\pi$
$824$	−8.64460e13	−6.53240
$825$	0	0
$826$	−1.98158e12	−0.148116
$827$	−7.56356e12	−0.562278	−0.281139	−	0.959667i	$-0.590712\pi$
$827$	−7.56356e12	−0.562278	−0.281139	+	0.959667i	$0.590712\pi$
$828$	−1.57460e13	−1.16421
$829$	1.86828e13	1.37387	0.686935	−	0.726719i	$-0.258956\pi$
$829$	1.86828e13	1.37387	0.686935	+	0.726719i	$0.258956\pi$
$830$	0	0
$831$	−7.41895e11	−0.0539682
$832$	3.77852e13	2.73380
$833$	−4.45880e12	−0.320860
$834$	2.24108e13	1.60402
$835$	0	0
$836$	9.82590e13	6.95735
$837$	2.68483e12	0.189083
$838$	−1.95336e13	−1.36831
$839$	−6.91387e12	−0.481717	−0.240859	−	0.970560i	$-0.577429\pi$
$839$	−6.91387e12	−0.481717	−0.240859	+	0.970560i	$0.577429\pi$
$840$	0	0
$841$	7.58972e12	0.523171
$842$	3.46426e13	2.37524
$843$	−7.60893e12	−0.518919
$844$	−5.67523e13	−3.84984
$845$	0	0
$846$	−5.22456e12	−0.350657
$847$	1.87548e13	1.25210
$848$	1.64827e13	1.09458
$849$	1.31369e13	0.867780
$850$	0	0
$851$	−7.76834e12	−0.507745
$852$	−4.67262e12	−0.303796
$853$	−2.71287e13	−1.75452	−0.877261	−	0.480013i	$-0.840632\pi$
$853$	−2.71287e13	−1.75452	−0.877261	+	0.480013i	$0.840632\pi$
$854$	5.98851e12	0.385264
$855$	0	0
$856$	−4.82538e13	−3.07184
$857$	−2.79535e12	−0.177020	−0.0885099	−	0.996075i	$-0.528210\pi$
$857$	−2.79535e12	−0.177020	−0.0885099	+	0.996075i	$0.528210\pi$
$858$	−1.78418e13	−1.12394
$859$	−3.62878e12	−0.227401	−0.113700	−	0.993515i	$-0.536270\pi$
$859$	−3.62878e12	−0.227401	−0.113700	+	0.993515i	$0.536270\pi$
$860$	0	0
$861$	−1.48092e12	−0.0918368
$862$	−1.24490e13	−0.767983
$863$	−2.59146e13	−1.59036	−0.795181	−	0.606372i	$-0.792624\pi$
$863$	−2.59146e13	−1.59036	−0.795181	+	0.606372i	$0.792624\pi$
$864$	−1.44524e13	−0.882323
$865$	0	0
$866$	−1.62885e13	−0.984128
$867$	−7.09087e12	−0.426200
$868$	2.83804e13	1.69699
$869$	9.90417e12	0.589154
$870$	0	0
$871$	8.90241e11	0.0524114
$872$	1.82835e13	1.07087
$873$	3.23732e12	0.188634
$874$	5.87005e13	3.40283
$875$	0	0
$876$	−8.24508e12	−0.473071
$877$	−2.03825e13	−1.16348	−0.581741	−	0.813374i	$-0.697628\pi$
$877$	−2.03825e13	−1.16348	−0.581741	+	0.813374i	$0.697628\pi$
$878$	2.46288e13	1.39868
$879$	2.93696e12	0.165939
$880$	0	0
$881$	−1.19114e13	−0.666146	−0.333073	−	0.942901i	$-0.608086\pi$
$881$	−1.19114e13	−0.666146	−0.333073	+	0.942901i	$0.608086\pi$
$882$	7.35070e12	0.408997
$883$	−8.21427e12	−0.454722	−0.227361	−	0.973811i	$-0.573010\pi$
$883$	−8.21427e12	−0.454722	−0.227361	+	0.973811i	$0.573010\pi$
$884$	1.49686e13	0.824417
$885$	0	0
$886$	−4.33341e13	−2.36253
$887$	2.54249e13	1.37912	0.689560	−	0.724228i	$-0.257804\pi$
$887$	2.54249e13	1.37912	0.689560	+	0.724228i	$0.257804\pi$
$888$	−1.57370e13	−0.849305
$889$	1.20685e13	0.648028
$890$	0	0
$891$	3.65078e12	0.194060
$892$	−4.58445e13	−2.42463
$893$	1.43895e13	0.757208
$894$	−2.15029e13	−1.12584
$895$	0	0
$896$	−5.66106e13	−2.93435
$897$	−7.87465e12	−0.406130
$898$	−1.88352e13	−0.966554
$899$	−2.37480e13	−1.21257
$900$	0	0
$901$	2.65579e12	0.134256
$902$	1.76968e13	0.890153
$903$	−5.20177e12	−0.260349
$904$	1.21417e13	0.604676
$905$	0	0
$906$	4.02720e11	0.0198576
$907$	−1.70360e13	−0.835864	−0.417932	−	0.908478i	$-0.637245\pi$
$907$	−1.70360e13	−0.835864	−0.417932	+	0.908478i	$0.637245\pi$
$908$	−3.26804e13	−1.59552
$909$	5.24248e12	0.254683
$910$	0	0
$911$	−1.12020e13	−0.538843	−0.269421	−	0.963022i	$-0.586832\pi$
$911$	−1.12020e13	−0.538843	−0.269421	+	0.963022i	$0.586832\pi$
$912$	7.08659e13	3.39204
$913$	6.52711e13	3.10887
$914$	−5.40967e13	−2.56397
$915$	0	0
$916$	−6.13252e13	−2.87812
$917$	1.35954e13	0.634935
$918$	−4.14577e12	−0.192669
$919$	−3.82553e13	−1.76918	−0.884589	−	0.466372i	$-0.845561\pi$
$919$	−3.82553e13	−1.76918	−0.884589	+	0.466372i	$0.845561\pi$
$920$	0	0
$921$	2.06343e13	0.944977
$922$	4.39844e13	2.00452
$923$	−2.33681e12	−0.105978
$924$	3.85911e13	1.74166
$925$	0	0
$926$	5.28629e13	2.36266
$927$	1.36843e13	0.608644
$928$	1.27835e14	5.65827
$929$	3.96522e13	1.74661	0.873307	−	0.487170i	$-0.161971\pi$
$929$	3.96522e13	1.74661	0.873307	+	0.487170i	$0.161971\pi$
$930$	0	0
$931$	−2.02454e13	−0.883187
$932$	−2.69132e13	−1.16841
$933$	2.01175e13	0.869172
$934$	2.63962e13	1.13496
$935$	0	0
$936$	−1.59524e13	−0.679335
$937$	5.22557e12	0.221465	0.110733	−	0.993850i	$-0.464680\pi$
$937$	5.22557e12	0.221465	0.110733	+	0.993850i	$0.464680\pi$
$938$	−2.60635e12	−0.109931
$939$	−1.67860e13	−0.704616
$940$	0	0
$941$	−1.63723e13	−0.680703	−0.340351	−	0.940298i	$-0.610546\pi$
$941$	−1.63723e13	−0.680703	−0.340351	+	0.940298i	$0.610546\pi$
$942$	3.04629e13	1.26050
$943$	7.81066e12	0.321651
$944$	1.26173e13	0.517120
$945$	0	0
$946$	6.21605e13	2.52351
$947$	−3.54022e13	−1.43039	−0.715195	−	0.698924i	$-0.753662\pi$
$947$	−3.54022e13	−1.43039	−0.715195	+	0.698924i	$0.753662\pi$
$948$	1.36985e13	0.550853
$949$	−4.12342e12	−0.165029
$950$	0	0
$951$	−4.01681e12	−0.159246
$952$	−2.83295e13	−1.11782
$953$	−2.06226e12	−0.0809888	−0.0404944	−	0.999180i	$-0.512893\pi$
$953$	−2.06226e12	−0.0809888	−0.0404944	+	0.999180i	$0.512893\pi$
$954$	−4.37830e12	−0.171134
$955$	0	0
$956$	1.44236e14	5.58487
$957$	−3.22920e13	−1.24449
$958$	−9.03600e13	−3.46602
$959$	−9.66033e11	−0.0368815
$960$	0	0
$961$	−9.17137e11	−0.0346880
$962$	−1.21745e13	−0.458315
$963$	7.63850e12	0.286213
$964$	−4.99006e12	−0.186105
$965$	0	0
$966$	2.30545e13	0.851842
$967$	3.58181e13	1.31730	0.658648	−	0.752451i	$-0.271129\pi$
$967$	3.58181e13	1.31730	0.658648	+	0.752451i	$0.271129\pi$
$968$	−2.00385e14	−7.33542
$969$	1.14183e13	0.416049
$970$	0	0
$971$	−4.16855e13	−1.50487	−0.752434	−	0.658667i	$-0.771120\pi$
$971$	−4.16855e13	−1.50487	−0.752434	+	0.658667i	$0.771120\pi$
$972$	5.04940e12	0.181444
$973$	−2.42421e13	−0.867085
$974$	8.70477e13	3.09914
$975$	0	0
$976$	−3.81304e13	−1.34508
$977$	−2.61832e13	−0.919384	−0.459692	−	0.888078i	$-0.652040\pi$
$977$	−2.61832e13	−0.919384	−0.459692	+	0.888078i	$0.652040\pi$
$978$	2.67485e13	0.934922
$979$	−8.11693e13	−2.82403
$980$	0	0
$981$	−2.89426e12	−0.0997762
$982$	5.50983e13	1.89076
$983$	3.59451e13	1.22786	0.613930	−	0.789360i	$-0.289588\pi$
$983$	3.59451e13	1.22786	0.613930	+	0.789360i	$0.289588\pi$
$984$	1.58227e13	0.538027
$985$	0	0
$986$	3.66703e13	1.23557
$987$	5.65147e12	0.189554
$988$	6.79657e13	2.26926
$989$	2.74352e13	0.911853
$990$	0	0
$991$	−2.19789e13	−0.723892	−0.361946	−	0.932199i	$-0.617888\pi$
$991$	−2.19789e13	−0.723892	−0.361946	+	0.932199i	$0.617888\pi$
$992$	−1.37387e14	−4.50447
$993$	−2.24200e13	−0.731751
$994$	6.84144e12	0.222284
$995$	0	0
$996$	9.02767e13	2.90676
$997$	4.20501e13	1.34784	0.673921	−	0.738804i	$-0.264609\pi$
$997$	4.20501e13	1.34784	0.673921	+	0.738804i	$0.264609\pi$
$998$	3.05516e13	0.974869
$999$	2.49115e12	0.0791325

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.10.a.j.1.1	✓	4
3.2	odd	2		225.10.a.t.1.4		4
5.2	odd	4		75.10.b.h.49.1		8
5.3	odd	4		75.10.b.h.49.8		8
5.4	even	2		75.10.a.k.1.4	yes	4
15.2	even	4		225.10.b.n.199.8		8
15.8	even	4		225.10.b.n.199.1		8
15.14	odd	2		225.10.a.r.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.10.a.j.1.1	✓	4	1.1	even	1	trivial
75.10.a.k.1.4	yes	4	5.4	even	2
75.10.b.h.49.1		8	5.2	odd	4
75.10.b.h.49.8		8	5.3	odd	4
225.10.a.r.1.1		4	15.14	odd	2
225.10.a.t.1.4		4	3.2	odd	2
225.10.b.n.199.1		8	15.8	even	4
225.10.b.n.199.8		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)

\(f(q)\)	\(=\)	\(q-44.2736 q^{2} +81.0000 q^{3} +1448.15 q^{4} -3586.16 q^{6} +3879.20 q^{7} -41447.0 q^{8} +6561.00 q^{9} +O(q^{10})\)	\(q-44.2736 q^{2} +81.0000 q^{3} +1448.15 q^{4} -3586.16 q^{6} +3879.20 q^{7} -41447.0 q^{8} +6561.00 q^{9} +84809.6 q^{11} +117301. q^{12} +58662.8 q^{13} -171746. q^{14} +1.09355e6 q^{16} +176199. q^{17} -290479. q^{18} +800041. q^{19} +314215. q^{21} -3.75483e6 q^{22} -1.65723e6 q^{23} -3.35720e6 q^{24} -2.59721e6 q^{26} +531441. q^{27} +5.61768e6 q^{28} -4.70073e6 q^{29} +5.05198e6 q^{31} -2.71947e7 q^{32} +6.86958e6 q^{33} -7.80099e6 q^{34} +9.50134e6 q^{36} +4.68753e6 q^{37} -3.54207e7 q^{38} +4.75168e6 q^{39} -4.71307e6 q^{41} -1.39114e7 q^{42} -1.65548e7 q^{43} +1.22817e8 q^{44} +7.33718e7 q^{46} +1.79860e7 q^{47} +8.85778e7 q^{48} -2.53054e7 q^{49} +1.42722e7 q^{51} +8.49527e7 q^{52} +1.50727e7 q^{53} -2.35288e7 q^{54} -1.60781e8 q^{56} +6.48033e7 q^{57} +2.08118e8 q^{58} +1.15379e7 q^{59} -3.48684e7 q^{61} -2.23669e8 q^{62} +2.54514e7 q^{63} +6.44110e8 q^{64} -3.04141e8 q^{66} +1.51756e7 q^{67} +2.55164e8 q^{68} -1.34236e8 q^{69} -3.98346e7 q^{71} -2.71934e8 q^{72} -7.02902e7 q^{73} -2.07534e8 q^{74} +1.15858e9 q^{76} +3.28993e8 q^{77} -2.10374e8 q^{78} +1.16781e8 q^{79} +4.30467e7 q^{81} +2.08665e8 q^{82} +7.69619e8 q^{83} +4.55032e8 q^{84} +7.32941e8 q^{86} -3.80759e8 q^{87} -3.51510e9 q^{88} -9.57076e8 q^{89} +2.27564e8 q^{91} -2.39993e9 q^{92} +4.09210e8 q^{93} -7.96305e8 q^{94} -2.20277e9 q^{96} +4.93418e8 q^{97} +1.12036e9 q^{98} +5.56436e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 324 q^{3} + 1792 q^{4} - 162 q^{6} + 13036 q^{7} - 24636 q^{8} + 26244 q^{9}+O(q^{10}) \) 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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