LMFDB - Embedded newform 750.6.c.a.499.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [750,6,Mod(499,750)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(750, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("750.499");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 750 = 2 \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	750.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$120.287864860$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1810x^{4} + 801025 $ x^8 + 1810*x^4 + 801025
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			499.7
Root			$-4.01473 - 4.01473i$ of defining polynomial
Character	$\chi$	$=$	750.499
Dual form			750.6.c.a.499.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} +15.0595i q^{7} -64.0000i q^{8} -81.0000 q^{9} +O(q^{10})$	\(q+4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} +15.0595i q^{7} -64.0000i q^{8} -81.0000 q^{9} +207.452 q^{11} -144.000i q^{12} +615.187i q^{13} -60.2380 q^{14} +256.000 q^{16} +25.1887i q^{17} -324.000i q^{18} +2055.04 q^{19} -135.535 q^{21} +829.808i q^{22} -4416.11i q^{23} +576.000 q^{24} -2460.75 q^{26} -729.000i q^{27} -240.952i q^{28} -5480.45 q^{29} -7370.78 q^{31} +1024.00i q^{32} +1867.07i q^{33} -100.755 q^{34} +1296.00 q^{36} -5754.71i q^{37} +8220.17i q^{38} -5536.69 q^{39} -5461.45 q^{41} -542.142i q^{42} +10123.1i q^{43} -3319.23 q^{44} +17664.4 q^{46} -6024.16i q^{47} +2304.00i q^{48} +16580.2 q^{49} -226.698 q^{51} -9843.00i q^{52} -27466.7i q^{53} +2916.00 q^{54} +963.808 q^{56} +18495.4i q^{57} -21921.8i q^{58} -28672.9 q^{59} -22423.7 q^{61} -29483.1i q^{62} -1219.82i q^{63} -4096.00 q^{64} -7468.27 q^{66} -7255.70i q^{67} -403.019i q^{68} +39745.0 q^{69} +4667.31 q^{71} +5184.00i q^{72} -76795.5i q^{73} +23018.8 q^{74} -32880.7 q^{76} +3124.12i q^{77} -22146.7i q^{78} -15294.0 q^{79} +6561.00 q^{81} -21845.8i q^{82} +15197.5i q^{83} +2168.57 q^{84} -40492.5 q^{86} -49324.1i q^{87} -13276.9i q^{88} +100719. q^{89} -9264.41 q^{91} +70657.7i q^{92} -66337.1i q^{93} +24096.6 q^{94} -9216.00 q^{96} -125075. i q^{97} +66320.8i q^{98} -16803.6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 128 q^{4} - 288 q^{6} - 648 q^{9}+O(q^{10}) $ 8 * q - 128 * q^4 - 288 * q^6 - 648 * q^9	$ 8 q - 128 q^{4} - 288 q^{6} - 648 q^{9} - 520 q^{11} + 688 q^{14} + 2048 q^{16} + 2672 q^{19} + 1548 q^{21} + 4608 q^{24} - 1216 q^{26} - 7072 q^{29} - 26636 q^{31} - 8592 q^{34} + 10368 q^{36} - 2736 q^{39} + 21172 q^{41} + 8320 q^{44} - 7680 q^{46} + 75868 q^{49} - 19332 q^{51} + 23328 q^{54} - 11008 q^{56} + 31180 q^{59} - 76620 q^{61} - 32768 q^{64} + 18720 q^{66} - 17280 q^{69} - 164820 q^{71} - 114944 q^{74} - 42752 q^{76} + 342456 q^{79} + 52488 q^{81} - 24768 q^{84} + 52432 q^{86} + 356856 q^{89} - 340324 q^{91} + 118720 q^{94} - 73728 q^{96} + 42120 q^{99}+O(q^{100}) $ 8 * q - 128 * q^4 - 288 * q^6 - 648 * q^9 - 520 * q^11 + 688 * q^14 + 2048 * q^16 + 2672 * q^19 + 1548 * q^21 + 4608 * q^24 - 1216 * q^26 - 7072 * q^29 - 26636 * q^31 - 8592 * q^34 + 10368 * q^36 - 2736 * q^39 + 21172 * q^41 + 8320 * q^44 - 7680 * q^46 + 75868 * q^49 - 19332 * q^51 + 23328 * q^54 - 11008 * q^56 + 31180 * q^59 - 76620 * q^61 - 32768 * q^64 + 18720 * q^66 - 17280 * q^69 - 164820 * q^71 - 114944 * q^74 - 42752 * q^76 + 342456 * q^79 + 52488 * q^81 - 24768 * q^84 + 52432 * q^86 + 356856 * q^89 - 340324 * q^91 + 118720 * q^94 - 73728 * q^96 + 42120 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/750\mathbb{Z}\right)^\times$.

$n$	$127$	$251$
$\chi(n)$	$-1$	$1$

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 $

$2$	4.00000i	0.707107i
$2$	4.00000i	0.707107i
$3$	9.00000i	0.577350i
$3$	9.00000i	0.577350i
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−36.0000	−0.408248
$7$	15.0595i	0.116162i	0.998312	+	0.0580812i	$0.0184982\pi$
$7$	15.0595i	0.116162i	−0.998312	+	0.0580812i	$0.981502\pi$
$8$	− 64.0000i	− 0.353553i
$9$	−81.0000	−0.333333
$10$	0	0
$11$	207.452	0.516935	0.258468	−	0.966020i	$-0.416783\pi$
$11$	207.452	0.516935	0.258468	+	0.966020i	$0.416783\pi$
$12$	− 144.000i	− 0.288675i
$13$	615.187i	1.00960i	0.863237	+	0.504800i	$0.168434\pi$
$13$	615.187i	1.00960i	−0.863237	+	0.504800i	$0.831566\pi$
$14$	−60.2380	−0.0821392
$15$	0	0
$16$	256.000	0.250000
$17$	25.1887i	0.0211389i	0.999944	+	0.0105695i	$0.00336443\pi$
$17$	25.1887i	0.0211389i	−0.999944	+	0.0105695i	$0.996636\pi$
$18$	− 324.000i	− 0.235702i
$19$	2055.04	1.30598	0.652991	−	0.757366i	$-0.273514\pi$
$19$	2055.04	1.30598	0.652991	+	0.757366i	$0.273514\pi$
$20$	0	0
$21$	−135.535	−0.0670663
$22$	829.808i	0.365528i
$23$	− 4416.11i	− 1.74068i	−0.492447	−	0.870342i	$-0.663898\pi$
$23$	− 4416.11i	− 1.74068i	0.492447	−	0.870342i	$-0.336102\pi$
$24$	576.000	0.204124
$25$	0	0
$26$	−2460.75	−0.713895
$27$	− 729.000i	− 0.192450i
$28$	− 240.952i	− 0.0580812i
$29$	−5480.45	−1.21010	−0.605050	−	0.796187i	$-0.706847\pi$
$29$	−5480.45	−1.21010	−0.605050	+	0.796187i	$0.706847\pi$
$30$	0	0
$31$	−7370.78	−1.37756	−0.688778	−	0.724972i	$-0.741853\pi$
$31$	−7370.78	−1.37756	−0.688778	+	0.724972i	$0.741853\pi$
$32$	1024.00i	0.176777i
$33$	1867.07i	0.298453i
$34$	−100.755	−0.0149475
$35$	0	0
$36$	1296.00	0.166667
$37$	− 5754.71i	− 0.691065i	−0.938407	−	0.345532i	$-0.887698\pi$
$37$	− 5754.71i	− 0.691065i	0.938407	−	0.345532i	$-0.112302\pi$
$38$	8220.17i	0.923468i
$39$	−5536.69	−0.582892
$40$	0	0
$41$	−5461.45	−0.507398	−0.253699	−	0.967283i	$-0.581647\pi$
$41$	−5461.45	−0.507398	−0.253699	+	0.967283i	$0.581647\pi$
$42$	− 542.142i	− 0.0474231i
$43$	10123.1i	0.834917i	0.908696	+	0.417459i	$0.137079\pi$
$43$	10123.1i	0.834917i	−0.908696	+	0.417459i	$0.862921\pi$
$44$	−3319.23	−0.258468
$45$	0	0
$46$	17664.4	1.23085
$47$	− 6024.16i	− 0.397788i	−0.980021	−	0.198894i	$-0.936265\pi$
$47$	− 6024.16i	− 0.397788i	0.980021	−	0.198894i	$-0.0637349\pi$
$48$	2304.00i	0.144338i
$49$	16580.2	0.986506
$50$	0	0
$51$	−226.698	−0.0122046
$52$	− 9843.00i	− 0.504800i
$53$	− 27466.7i	− 1.34312i	−0.740948	−	0.671562i	$-0.765624\pi$
$53$	− 27466.7i	− 1.34312i	0.740948	−	0.671562i	$-0.234376\pi$
$54$	2916.00	0.136083
$55$	0	0
$56$	963.808	0.0410696
$57$	18495.4i	0.754009i
$58$	− 21921.8i	− 0.855670i
$59$	−28672.9	−1.07236	−0.536180	−	0.844103i	$-0.680133\pi$
$59$	−28672.9	−1.07236	−0.536180	+	0.844103i	$0.680133\pi$
$60$	0	0
$61$	−22423.7	−0.771584	−0.385792	−	0.922586i	$-0.626072\pi$
$61$	−22423.7	−0.771584	−0.385792	+	0.922586i	$0.626072\pi$
$62$	− 29483.1i	− 0.974080i
$63$	− 1219.82i	− 0.0387208i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	−7468.27	−0.211038
$67$	− 7255.70i	− 0.197466i	−0.995114	−	0.0987330i	$-0.968521\pi$
$67$	− 7255.70i	− 0.197466i	0.995114	−	0.0987330i	$-0.0314790\pi$
$68$	− 403.019i	− 0.0105695i
$69$	39745.0	1.00498
$70$	0	0
$71$	4667.31	0.109881	0.0549403	−	0.998490i	$-0.482503\pi$
$71$	4667.31	0.109881	0.0549403	+	0.998490i	$0.482503\pi$
$72$	5184.00i	0.117851i
$73$	− 76795.5i	− 1.68666i	−0.537393	−	0.843332i	$-0.680591\pi$
$73$	− 76795.5i	− 1.68666i	0.537393	−	0.843332i	$-0.319409\pi$
$74$	23018.8	0.488657
$75$	0	0
$76$	−32880.7	−0.652991
$77$	3124.12i	0.0600484i
$78$	− 22146.7i	− 0.412167i
$79$	−15294.0	−0.275711	−0.137855	−	0.990452i	$-0.544021\pi$
$79$	−15294.0	−0.275711	−0.137855	+	0.990452i	$0.544021\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	− 21845.8i	− 0.358784i
$83$	15197.5i	0.242146i	0.992644	+	0.121073i	$0.0386335\pi$
$83$	15197.5i	0.242146i	−0.992644	+	0.121073i	$0.961367\pi$
$84$	2168.57	0.0335332
$85$	0	0
$86$	−40492.5	−0.590376
$87$	− 49324.1i	− 0.698652i
$88$	− 13276.9i	− 0.182764i
$89$	100719.	1.34783	0.673917	−	0.738807i	$-0.264610\pi$
$89$	100719.	1.34783	0.673917	+	0.738807i	$0.264610\pi$
$90$	0	0
$91$	−9264.41	−0.117277
$92$	70657.7i	0.870342i
$93$	− 66337.1i	− 0.795333i
$94$	24096.6	0.281279
$95$	0	0
$96$	−9216.00	−0.102062
$97$	− 125075.i	− 1.34972i	−0.737948	−	0.674858i	$-0.764205\pi$
$97$	− 125075.i	− 1.34972i	0.737948	−	0.674858i	$-0.235795\pi$
$98$	66320.8i	0.697565i
$99$	−16803.6	−0.172312
$100$	0	0
$101$	53077.5	0.517734	0.258867	−	0.965913i	$-0.416651\pi$
$101$	53077.5	0.517734	0.258867	+	0.965913i	$0.416651\pi$
$102$	− 906.793i	− 0.00862994i
$103$	115200.i	1.06994i	0.844870	+	0.534971i	$0.179678\pi$
$103$	115200.i	1.06994i	−0.844870	+	0.534971i	$0.820322\pi$
$104$	39372.0	0.356947
$105$	0	0
$106$	109867.	0.949732
$107$	− 93751.9i	− 0.791627i	−0.918331	−	0.395813i	$-0.870463\pi$
$107$	− 93751.9i	− 0.791627i	0.918331	−	0.395813i	$-0.129537\pi$
$108$	11664.0i	0.0962250i
$109$	192306.	1.55034	0.775168	−	0.631756i	$-0.217665\pi$
$109$	192306.	1.55034	0.775168	+	0.631756i	$0.217665\pi$
$110$	0	0
$111$	51792.3	0.398986
$112$	3855.23i	0.0290406i
$113$	4670.05i	0.0344053i	0.999852	+	0.0172027i	$0.00547605\pi$
$113$	4670.05i	0.0344053i	−0.999852	+	0.0172027i	$0.994524\pi$
$114$	−73981.6	−0.533165
$115$	0	0
$116$	87687.3	0.605050
$117$	− 49830.2i	− 0.336533i
$118$	− 114691.i	− 0.758273i
$119$	−379.329	−0.00245555
$120$	0	0
$121$	−118015.	−0.732778
$122$	− 89694.9i	− 0.545592i
$123$	− 49153.1i	− 0.292946i
$124$	117933.	0.688778
$125$	0	0
$126$	4879.28	0.0273797
$127$	52582.7i	0.289290i	0.989484	+	0.144645i	$0.0462040\pi$
$127$	52582.7i	0.289290i	−0.989484	+	0.144645i	$0.953796\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	−91108.1	−0.482040
$130$	0	0
$131$	−287789.	−1.46520	−0.732598	−	0.680661i	$-0.761693\pi$
$131$	−287789.	−1.46520	−0.732598	+	0.680661i	$0.761693\pi$
$132$	− 29873.1i	− 0.149226i
$133$	30947.9i	0.151706i
$134$	29022.8	0.139629
$135$	0	0
$136$	1612.08	0.00747374
$137$	− 323964.i	− 1.47467i	−0.675525	−	0.737337i	$-0.736083\pi$
$137$	− 323964.i	− 1.47467i	0.675525	−	0.737337i	$-0.263917\pi$
$138$	158980.i	0.710631i
$139$	93215.3	0.409214	0.204607	−	0.978844i	$-0.434408\pi$
$139$	93215.3	0.409214	0.204607	+	0.978844i	$0.434408\pi$
$140$	0	0
$141$	54217.4	0.229663
$142$	18669.2i	0.0776973i
$143$	127622.i	0.521897i
$144$	−20736.0	−0.0833333
$145$	0	0
$146$	307182.	1.19265
$147$	149222.i	0.569560i
$148$	92075.3i	0.345532i
$149$	304482.	1.12356	0.561780	−	0.827287i	$-0.310117\pi$
$149$	304482.	1.12356	0.561780	+	0.827287i	$0.310117\pi$
$150$	0	0
$151$	150020.	0.535434	0.267717	−	0.963498i	$-0.413731\pi$
$151$	150020.	0.535434	0.267717	+	0.963498i	$0.413731\pi$
$152$	− 131523.i	− 0.461734i
$153$	− 2040.28i	− 0.00704631i
$154$	−12496.5	−0.0424606
$155$	0	0
$156$	88587.0	0.291446
$157$	20217.1i	0.0654592i	0.999464	+	0.0327296i	$0.0104200\pi$
$157$	20217.1i	0.0654592i	−0.999464	+	0.0327296i	$0.989580\pi$
$158$	− 61176.0i	− 0.194957i
$159$	247200.	0.775453
$160$	0	0
$161$	66504.3	0.202202
$162$	26244.0i	0.0785674i
$163$	− 388948.i	− 1.14663i	−0.819335	−	0.573314i	$-0.805657\pi$
$163$	− 388948.i	− 1.14663i	0.819335	−	0.573314i	$-0.194343\pi$
$164$	87383.3	0.253699
$165$	0	0
$166$	−60790.0	−0.171223
$167$	40159.0i	0.111427i	0.998447	+	0.0557137i	$0.0177434\pi$
$167$	40159.0i	0.111427i	−0.998447	+	0.0557137i	$0.982257\pi$
$168$	8674.27i	0.0237115i
$169$	−7162.51	−0.0192907
$170$	0	0
$171$	−166459.	−0.435327
$172$	− 161970.i	− 0.417459i
$173$	218877.i	0.556012i	0.960579	+	0.278006i	$0.0896735\pi$
$173$	218877.i	0.556012i	−0.960579	+	0.278006i	$0.910326\pi$
$174$	197296.	0.494022
$175$	0	0
$176$	53107.7	0.129234
$177$	− 258056.i	− 0.619128i
$178$	402876.i	0.953063i
$179$	348289.	0.812471	0.406235	−	0.913769i	$-0.366841\pi$
$179$	348289.	0.812471	0.406235	+	0.913769i	$0.366841\pi$
$180$	0	0
$181$	−247462.	−0.561450	−0.280725	−	0.959788i	$-0.590575\pi$
$181$	−247462.	−0.561450	−0.280725	+	0.959788i	$0.590575\pi$
$182$	− 37057.6i	− 0.0829276i
$183$	− 201813.i	− 0.445474i
$184$	−282631.	−0.615425
$185$	0	0
$186$	265348.	0.562385
$187$	5225.44i	0.0109275i
$188$	96386.5i	0.198894i
$189$	10978.4	0.0223554
$190$	0	0
$191$	604496.	1.19897	0.599487	−	0.800384i	$-0.295371\pi$
$191$	604496.	1.19897	0.599487	+	0.800384i	$0.295371\pi$
$192$	− 36864.0i	− 0.0721688i
$193$	− 211625.i	− 0.408953i	−0.978871	−	0.204477i	$-0.934451\pi$
$193$	− 211625.i	− 0.408953i	0.978871	−	0.204477i	$-0.0655493\pi$
$194$	500301.	0.954393
$195$	0	0
$196$	−265283.	−0.493253
$197$	495102.i	0.908928i	0.890765	+	0.454464i	$0.150169\pi$
$197$	495102.i	0.908928i	−0.890765	+	0.454464i	$0.849831\pi$
$198$	− 67214.5i	− 0.121843i
$199$	770648.	1.37950	0.689752	−	0.724045i	$-0.257719\pi$
$199$	770648.	1.37950	0.689752	+	0.724045i	$0.257719\pi$
$200$	0	0
$201$	65301.3	0.114007
$202$	212310.i	0.366094i
$203$	− 82532.9i	− 0.140568i
$204$	3627.17	0.00610229
$205$	0	0
$206$	−460801.	−0.756564
$207$	357705.i	0.580228i
$208$	157488.i	0.252400i
$209$	426323.	0.675108
$210$	0	0
$211$	−147431.	−0.227972	−0.113986	−	0.993482i	$-0.536362\pi$
$211$	−147431.	−0.227972	−0.113986	+	0.993482i	$0.536362\pi$
$212$	439467.i	0.671562i
$213$	42005.8i	0.0634396i
$214$	375007.	0.559765
$215$	0	0
$216$	−46656.0	−0.0680414
$217$	− 111000.i	− 0.160020i
$218$	769222.i	1.09625i
$219$	691159.	0.973796
$220$	0	0
$221$	−15495.8	−0.0213419
$222$	207169.i	0.282126i
$223$	798238.i	1.07491i	0.843294	+	0.537453i	$0.180613\pi$
$223$	798238.i	1.07491i	−0.843294	+	0.537453i	$0.819387\pi$
$224$	−15420.9	−0.0205348
$225$	0	0
$226$	−18680.2	−0.0243282
$227$	− 1.16286e6i	− 1.49782i	−0.662669	−	0.748912i	$-0.730576\pi$
$227$	− 1.16286e6i	− 1.49782i	0.662669	−	0.748912i	$-0.269424\pi$
$228$	− 295926.i	− 0.377004i
$229$	953922.	1.20205	0.601027	−	0.799228i	$-0.294758\pi$
$229$	953922.	1.20205	0.601027	+	0.799228i	$0.294758\pi$
$230$	0	0
$231$	−28117.1	−0.0346689
$232$	350749.i	0.427835i
$233$	628239.i	0.758115i	0.925373	+	0.379058i	$0.123752\pi$
$233$	628239.i	0.758115i	−0.925373	+	0.379058i	$0.876248\pi$
$234$	199321.	0.237965
$235$	0	0
$236$	458766.	0.536180
$237$	− 137646.i	− 0.159182i
$238$	− 1517.32i	− 0.00173633i
$239$	290012.	0.328414	0.164207	−	0.986426i	$-0.447494\pi$
$239$	290012.	0.328414	0.164207	+	0.986426i	$0.447494\pi$
$240$	0	0
$241$	−1.76901e6	−1.96194	−0.980972	−	0.194150i	$-0.937805\pi$
$241$	−1.76901e6	−1.96194	−0.980972	+	0.194150i	$0.937805\pi$
$242$	− 472059.i	− 0.518152i
$243$	59049.0i	0.0641500i
$244$	358779.	0.385792
$245$	0	0
$246$	196612.	0.207144
$247$	1.26424e6i	1.31852i
$248$	471730.i	0.487040i
$249$	−136777.	−0.139803
$250$	0	0
$251$	−1.26262e6	−1.26500	−0.632499	−	0.774561i	$-0.717971\pi$
$251$	−1.26262e6	−1.26500	−0.632499	+	0.774561i	$0.717971\pi$
$252$	19517.1i	0.0193604i
$253$	− 916130.i	− 0.899821i
$254$	−210331.	−0.204559
$255$	0	0
$256$	65536.0	0.0625000
$257$	923033.i	0.871735i	0.900011	+	0.435867i	$0.143558\pi$
$257$	923033.i	0.871735i	−0.900011	+	0.435867i	$0.856442\pi$
$258$	− 364432.i	− 0.340854i
$259$	86662.9	0.0802757
$260$	0	0
$261$	443917.	0.403367
$262$	− 1.15116e6i	− 1.03605i
$263$	− 546009.i	− 0.486755i	−0.969932	−	0.243377i	$-0.921745\pi$
$263$	− 546009.i	− 0.486755i	0.969932	−	0.243377i	$-0.0782554\pi$
$264$	119492.	0.105519
$265$	0	0
$266$	−123792.	−0.107272
$267$	906471.i	0.778173i
$268$	116091.i	0.0987330i
$269$	593220.	0.499845	0.249922	−	0.968266i	$-0.419595\pi$
$269$	593220.	0.499845	0.249922	+	0.968266i	$0.419595\pi$
$270$	0	0
$271$	113240.	0.0936649	0.0468325	−	0.998903i	$-0.485087\pi$
$271$	113240.	0.0936649	0.0468325	+	0.998903i	$0.485087\pi$
$272$	6448.30i	0.00528473i
$273$	− 83379.7i	− 0.0677101i
$274$	1.29586e6	1.04275
$275$	0	0
$276$	−635920.	−0.502492
$277$	1.56779e6i	1.22769i	0.789428	+	0.613843i	$0.210377\pi$
$277$	1.56779e6i	1.22769i	−0.789428	+	0.613843i	$0.789623\pi$
$278$	372861.i	0.289358i
$279$	597034.	0.459186
$280$	0	0
$281$	1.72342e6	1.30204	0.651021	−	0.759060i	$-0.274341\pi$
$281$	1.72342e6	1.30204	0.651021	+	0.759060i	$0.274341\pi$
$282$	216870.i	0.162396i
$283$	− 2.44081e6i	− 1.81162i	−0.423684	−	0.905810i	$-0.639263\pi$
$283$	− 2.44081e6i	− 1.81162i	0.423684	−	0.905810i	$-0.360737\pi$
$284$	−74677.0	−0.0549403
$285$	0	0
$286$	−510487.	−0.369037
$287$	− 82246.7i	− 0.0589405i
$288$	− 82944.0i	− 0.0589256i
$289$	1.41922e6	0.999553
$290$	0	0
$291$	1.12568e6	0.779258
$292$	1.22873e6i	0.843332i
$293$	− 2.19256e6i	− 1.49205i	−0.665918	−	0.746025i	$-0.731960\pi$
$293$	− 2.19256e6i	− 1.49205i	0.665918	−	0.746025i	$-0.268040\pi$
$294$	−596888.	−0.402740
$295$	0	0
$296$	−368301.	−0.244328
$297$	− 151233.i	− 0.0994842i
$298$	1.21793e6i	0.794476i
$299$	2.71673e6	1.75739
$300$	0	0
$301$	−152449.	−0.0969859
$302$	600078.i	0.378609i
$303$	477698.i	0.298914i
$304$	526091.	0.326495
$305$	0	0
$306$	8161.13	0.00498250
$307$	− 1.83111e6i	− 1.10884i	−0.832236	−	0.554421i	$-0.812940\pi$
$307$	− 1.83111e6i	− 1.10884i	0.832236	−	0.554421i	$-0.187060\pi$
$308$	− 49986.0i	− 0.0300242i
$309$	−1.03680e6	−0.617732
$310$	0	0
$311$	1.98155e6	1.16172	0.580862	−	0.814002i	$-0.302715\pi$
$311$	1.98155e6	1.16172	0.580862	+	0.814002i	$0.302715\pi$
$312$	354348.i	0.206084i
$313$	− 261898.i	− 0.151102i	−0.997142	−	0.0755511i	$-0.975928\pi$
$313$	− 261898.i	− 0.151102i	0.997142	−	0.0755511i	$-0.0240716\pi$
$314$	−80868.5	−0.0462866
$315$	0	0
$316$	244704.	0.137855
$317$	536763.i	0.300009i	0.988685	+	0.150004i	$0.0479288\pi$
$317$	536763.i	0.300009i	−0.988685	+	0.150004i	$0.952071\pi$
$318$	988800.i	0.548328i
$319$	−1.13693e6	−0.625543
$320$	0	0
$321$	843767.	0.457046
$322$	266017.i	0.142978i
$323$	51763.8i	0.0276071i
$324$	−104976.	−0.0555556
$325$	0	0
$326$	1.55579e6	0.810789
$327$	1.73075e6i	0.895086i
$328$	349533.i	0.179392i
$329$	90720.8	0.0462080
$330$	0	0
$331$	−3.09171e6	−1.55106	−0.775530	−	0.631311i	$-0.782517\pi$
$331$	−3.09171e6	−1.55106	−0.775530	+	0.631311i	$0.782517\pi$
$332$	− 243160.i	− 0.121073i
$333$	466131.i	0.230355i
$334$	−160636.	−0.0787911
$335$	0	0
$336$	−34697.1	−0.0167666
$337$	400700.i	0.192196i	0.995372	+	0.0960980i	$0.0306362\pi$
$337$	400700.i	0.192196i	−0.995372	+	0.0960980i	$0.969364\pi$
$338$	− 28650.1i	− 0.0136406i
$339$	−42030.5	−0.0198639
$340$	0	0
$341$	−1.52908e6	−0.712107
$342$	− 665834.i	− 0.307823i
$343$	502795.i	0.230757i
$344$	647880.	0.295188
$345$	0	0
$346$	−875507.	−0.393160
$347$	− 1.05427e6i	− 0.470035i	−0.971991	−	0.235017i	$-0.924485\pi$
$347$	− 1.05427e6i	− 0.470035i	0.971991	−	0.235017i	$-0.0755147\pi$
$348$	789185.i	0.349326i
$349$	304984.	0.134034	0.0670168	−	0.997752i	$-0.478652\pi$
$349$	304984.	0.134034	0.0670168	+	0.997752i	$0.478652\pi$
$350$	0	0
$351$	448472.	0.194297
$352$	212431.i	0.0913821i
$353$	− 3.07212e6i	− 1.31221i	−0.754672	−	0.656103i	$-0.772204\pi$
$353$	− 3.07212e6i	− 1.31221i	0.754672	−	0.656103i	$-0.227796\pi$
$354$	1.03222e6	0.437789
$355$	0	0
$356$	−1.61150e6	−0.673917
$357$	− 3413.96i	− 0.00141771i
$358$	1.39316e6i	0.574503i
$359$	2.38760e6	0.977745	0.488873	−	0.872355i	$-0.337408\pi$
$359$	2.38760e6	0.977745	0.488873	+	0.872355i	$0.337408\pi$
$360$	0	0
$361$	1.74711e6	0.705588
$362$	− 989847.i	− 0.397005i
$363$	− 1.06213e6i	− 0.423070i
$364$	148231.	0.0586387
$365$	0	0
$366$	807254.	0.314998
$367$	1.55592e6i	0.603008i	0.953465	+	0.301504i	$0.0974886\pi$
$367$	1.55592e6i	0.603008i	−0.953465	+	0.301504i	$0.902511\pi$
$368$	− 1.13052e6i	− 0.435171i
$369$	442378.	0.169133
$370$	0	0
$371$	413634.	0.156020
$372$	1.06139e6i	0.397666i
$373$	− 194934.i	− 0.0725462i	−0.999342	−	0.0362731i	$-0.988451\pi$
$373$	− 194934.i	− 0.0725462i	0.999342	−	0.0362731i	$-0.0115486\pi$
$374$	−20901.8	−0.00772688
$375$	0	0
$376$	−385546.	−0.140639
$377$	− 3.37151e6i	− 1.22172i
$378$	43913.5i	0.0158077i
$379$	−1.19937e6	−0.428900	−0.214450	−	0.976735i	$-0.568796\pi$
$379$	−1.19937e6	−0.428900	−0.214450	+	0.976735i	$0.568796\pi$
$380$	0	0
$381$	−473244.	−0.167022
$382$	2.41798e6i	0.847803i
$383$	− 3.11967e6i	− 1.08671i	−0.839504	−	0.543353i	$-0.817155\pi$
$383$	− 3.11967e6i	− 1.08671i	0.839504	−	0.543353i	$-0.182845\pi$
$384$	147456.	0.0510310
$385$	0	0
$386$	846500.	0.289174
$387$	− 819973.i	− 0.278306i
$388$	2.00120e6i	0.674858i
$389$	−3.68995e6	−1.23636	−0.618182	−	0.786035i	$-0.712131\pi$
$389$	−3.68995e6	−1.23636	−0.618182	+	0.786035i	$0.712131\pi$
$390$	0	0
$391$	111236.	0.0367962
$392$	− 1.06113e6i	− 0.348783i
$393$	− 2.59010e6i	− 0.845932i
$394$	−1.98041e6	−0.642709
$395$	0	0
$396$	268858.	0.0861558
$397$	2.46138e6i	0.783794i	0.920009	+	0.391897i	$0.128181\pi$
$397$	2.46138e6i	0.783794i	−0.920009	+	0.391897i	$0.871819\pi$
$398$	3.08259e6i	0.975457i
$399$	−278531.	−0.0875874
$400$	0	0
$401$	4.67204e6	1.45093	0.725464	−	0.688261i	$-0.241625\pi$
$401$	4.67204e6	1.45093	0.725464	+	0.688261i	$0.241625\pi$
$402$	261205.i	0.0806151i
$403$	− 4.53441e6i	− 1.39078i
$404$	−849240.	−0.258867
$405$	0	0
$406$	330131.	0.0993967
$407$	− 1.19383e6i	− 0.357236i
$408$	14508.7i	0.00431497i
$409$	−2.35815e6	−0.697048	−0.348524	−	0.937300i	$-0.613317\pi$
$409$	−2.35815e6	−0.697048	−0.348524	+	0.937300i	$0.613317\pi$
$410$	0	0
$411$	2.91568e6	0.851403
$412$	− 1.84321e6i	− 0.534971i
$413$	− 431799.i	− 0.124568i
$414$	−1.43082e6	−0.410283
$415$	0	0
$416$	−629952.	−0.178474
$417$	838937.i	0.236260i
$418$	1.70529e6i	0.477373i
$419$	527492.	0.146785	0.0733923	−	0.997303i	$-0.476617\pi$
$419$	527492.	0.146785	0.0733923	+	0.997303i	$0.476617\pi$
$420$	0	0
$421$	−2.75474e6	−0.757487	−0.378744	−	0.925502i	$-0.623644\pi$
$421$	−2.75474e6	−0.757487	−0.378744	+	0.925502i	$0.623644\pi$
$422$	− 589723.i	− 0.161201i
$423$	487957.i	0.132596i
$424$	−1.75787e6	−0.474866
$425$	0	0
$426$	−168023.	−0.0448585
$427$	− 337690.i	− 0.0896289i
$428$	1.50003e6i	0.395813i
$429$	−1.14860e6	−0.301318
$430$	0	0
$431$	4.54408e6	1.17829	0.589146	−	0.808027i	$-0.299464\pi$
$431$	4.54408e6	1.17829	0.589146	+	0.808027i	$0.299464\pi$
$432$	− 186624.i	− 0.0481125i
$433$	− 1.18003e6i	− 0.302465i	−0.988498	−	0.151232i	$-0.951676\pi$
$433$	− 1.18003e6i	− 0.302465i	0.988498	−	0.151232i	$-0.0483242\pi$
$434$	444001.	0.113151
$435$	0	0
$436$	−3.07689e6	−0.775168
$437$	− 9.07529e6i	− 2.27330i
$438$	2.76464e6i	0.688578i
$439$	3.62542e6	0.897836	0.448918	−	0.893573i	$-0.351809\pi$
$439$	3.62542e6	0.897836	0.448918	+	0.893573i	$0.351809\pi$
$440$	0	0
$441$	−1.34300e6	−0.328835
$442$	− 61983.0i	− 0.0150910i
$443$	1.27872e6i	0.309574i	0.987948	+	0.154787i	$0.0494692\pi$
$443$	1.27872e6i	0.309574i	−0.987948	+	0.154787i	$0.950531\pi$
$444$	−828678.	−0.199493
$445$	0	0
$446$	−3.19295e6	−0.760073
$447$	2.74034e6i	0.648687i
$448$	− 61683.7i	− 0.0145203i
$449$	−3.25975e6	−0.763077	−0.381538	−	0.924353i	$-0.624606\pi$
$449$	−3.25975e6	−0.763077	−0.381538	+	0.924353i	$0.624606\pi$
$450$	0	0
$451$	−1.13299e6	−0.262292
$452$	− 74720.9i	− 0.0172027i
$453$	1.35018e6i	0.309133i
$454$	4.65142e6	1.05912
$455$	0	0
$456$	1.18371e6	0.266582
$457$	7.14653e6i	1.60068i	0.599546	+	0.800341i	$0.295348\pi$
$457$	7.14653e6i	1.60068i	−0.599546	+	0.800341i	$0.704652\pi$
$458$	3.81569e6i	0.849981i
$459$	18362.6	0.00406819
$460$	0	0
$461$	−3.96935e6	−0.869895	−0.434947	−	0.900456i	$-0.643233\pi$
$461$	−3.96935e6	−0.869895	−0.434947	+	0.900456i	$0.643233\pi$
$462$	− 112468.i	− 0.0245146i
$463$	3.02763e6i	0.656371i	0.944613	+	0.328186i	$0.106437\pi$
$463$	3.02763e6i	0.656371i	−0.944613	+	0.328186i	$0.893563\pi$
$464$	−1.40300e6	−0.302525
$465$	0	0
$466$	−2.51296e6	−0.536069
$467$	− 3.88702e6i	− 0.824754i	−0.911013	−	0.412377i	$-0.864699\pi$
$467$	− 3.88702e6i	− 0.824754i	0.911013	−	0.412377i	$-0.135301\pi$
$468$	797283.i	0.168267i
$469$	109267.	0.0229381
$470$	0	0
$471$	−181954.	−0.0377929
$472$	1.83506e6i	0.379137i
$473$	2.10006e6i	0.431598i
$474$	550584.	0.112558
$475$	0	0
$476$	6069.26	0.00122777
$477$	2.22480e6i	0.447708i
$478$	1.16005e6i	0.232224i
$479$	−5.38838e6	−1.07305	−0.536524	−	0.843885i	$-0.680263\pi$
$479$	−5.38838e6	−1.07305	−0.536524	+	0.843885i	$0.680263\pi$
$480$	0	0
$481$	3.54022e6	0.697699
$482$	− 7.07602e6i	− 1.38730i
$483$	598539.i	0.116741i
$484$	1.88823e6	0.366389
$485$	0	0
$486$	−236196.	−0.0453609
$487$	− 8.37200e6i	− 1.59958i	−0.600278	−	0.799791i	$-0.704944\pi$
$487$	− 8.37200e6i	− 1.59958i	0.600278	−	0.799791i	$-0.295056\pi$
$488$	1.43512e6i	0.272796i
$489$	3.50053e6	0.662007
$490$	0	0
$491$	−3.46979e6	−0.649530	−0.324765	−	0.945795i	$-0.605285\pi$
$491$	−3.46979e6	−0.649530	−0.324765	+	0.945795i	$0.605285\pi$
$492$	786449.i	0.146473i
$493$	− 138045.i	− 0.0255802i
$494$	−5.05695e6	−0.932333
$495$	0	0
$496$	−1.88692e6	−0.344389
$497$	70287.3i	0.0127640i
$498$	− 547110.i	− 0.0988556i
$499$	−1.50541e6	−0.270647	−0.135323	−	0.990801i	$-0.543207\pi$
$499$	−1.50541e6	−0.270647	−0.135323	+	0.990801i	$0.543207\pi$
$500$	0	0
$501$	−361431.	−0.0643326
$502$	− 5.05050e6i	− 0.894489i
$503$	− 6.65800e6i	− 1.17334i	−0.809826	−	0.586670i	$-0.800439\pi$
$503$	− 6.65800e6i	− 1.17334i	0.809826	−	0.586670i	$-0.199561\pi$
$504$	−78068.4	−0.0136899
$505$	0	0
$506$	3.66452e6	0.636269
$507$	− 64462.6i	− 0.0111375i
$508$	− 841323.i	− 0.144645i
$509$	−5.79333e6	−0.991137	−0.495568	−	0.868569i	$-0.665040\pi$
$509$	−5.79333e6	−0.991137	−0.495568	+	0.868569i	$0.665040\pi$
$510$	0	0
$511$	1.15650e6	0.195927
$512$	262144.i	0.0441942i
$513$	− 1.49813e6i	− 0.251336i
$514$	−3.69213e6	−0.616410
$515$	0	0
$516$	1.45773e6	0.241020
$517$	− 1.24972e6i	− 0.205631i
$518$	346652.i	0.0567635i
$519$	−1.96989e6	−0.321014
$520$	0	0
$521$	2.98946e6	0.482502	0.241251	−	0.970463i	$-0.422442\pi$
$521$	2.98946e6	0.482502	0.241251	+	0.970463i	$0.422442\pi$
$522$	1.77567e6i	0.285223i
$523$	7.50145e6i	1.19920i	0.800301	+	0.599599i	$0.204673\pi$
$523$	7.50145e6i	1.19920i	−0.800301	+	0.599599i	$0.795327\pi$
$524$	4.60462e6	0.732598
$525$	0	0
$526$	2.18404e6	0.344188
$527$	− 185660.i	− 0.0291201i
$528$	477969.i	0.0746131i
$529$	−1.30657e7	−2.02998
$530$	0	0
$531$	2.32250e6	0.357454
$532$	− 495167.i	− 0.0758529i
$533$	− 3.35982e6i	− 0.512268i
$534$	−3.62589e6	−0.550251
$535$	0	0
$536$	−464365.	−0.0698147
$537$	3.13460e6i	0.469080i
$538$	2.37288e6i	0.353444i
$539$	3.43960e6	0.509960
$540$	0	0
$541$	−2.61615e6	−0.384299	−0.192149	−	0.981366i	$-0.561546\pi$
$541$	−2.61615e6	−0.384299	−0.192149	+	0.981366i	$0.561546\pi$
$542$	452960.i	0.0662311i
$543$	− 2.22715e6i	− 0.324154i
$544$	−25793.2	−0.00373687
$545$	0	0
$546$	333519.	0.0478783
$547$	− 1.44019e6i	− 0.205803i	−0.994692	−	0.102902i	$-0.967187\pi$
$547$	− 1.44019e6i	− 0.205803i	0.994692	−	0.102902i	$-0.0328127\pi$
$548$	5.18343e6i	0.737337i
$549$	1.81632e6	0.257195
$550$	0	0
$551$	−1.12626e7	−1.58037
$552$	− 2.54368e6i	− 0.355316i
$553$	− 230320.i	− 0.0320272i
$554$	−6.27114e6	−0.868105
$555$	0	0
$556$	−1.49144e6	−0.204607
$557$	− 1.06758e7i	− 1.45801i	−0.684508	−	0.729005i	$-0.739983\pi$
$557$	− 1.06758e7i	− 1.45801i	0.684508	−	0.729005i	$-0.260017\pi$
$558$	2.38813e6i	0.324693i
$559$	−6.22762e6	−0.842932
$560$	0	0
$561$	−47029.0	−0.00630897
$562$	6.89367e6i	0.920682i
$563$	− 8.80097e6i	− 1.17020i	−0.810962	−	0.585099i	$-0.801055\pi$
$563$	− 8.80097e6i	− 1.17020i	0.810962	−	0.585099i	$-0.198945\pi$
$564$	−867479.	−0.114832
$565$	0	0
$566$	9.76322e6	1.28101
$567$	98805.3i	0.0129069i
$568$	− 298708.i	− 0.0388486i
$569$	−7.44140e6	−0.963549	−0.481775	−	0.876295i	$-0.660008\pi$
$569$	−7.44140e6	−0.963549	−0.481775	+	0.876295i	$0.660008\pi$
$570$	0	0
$571$	−6.84316e6	−0.878348	−0.439174	−	0.898402i	$-0.644729\pi$
$571$	−6.84316e6	−0.878348	−0.439174	+	0.898402i	$0.644729\pi$
$572$	− 2.04195e6i	− 0.260949i
$573$	5.44047e6i	0.692228i
$574$	328987.	0.0416772
$575$	0	0
$576$	331776.	0.0416667
$577$	6.73079e6i	0.841640i	0.907144	+	0.420820i	$0.138258\pi$
$577$	6.73079e6i	0.841640i	−0.907144	+	0.420820i	$0.861742\pi$
$578$	5.67689e6i	0.706791i
$579$	1.90463e6	0.236109
$580$	0	0
$581$	−228867.	−0.0281282
$582$	4.50271e6i	0.551019i
$583$	− 5.69801e6i	− 0.694308i
$584$	−4.91491e6	−0.596326
$585$	0	0
$586$	8.77026e6	1.05504
$587$	− 1.24111e7i	− 1.48667i	−0.668919	−	0.743335i	$-0.733243\pi$
$587$	− 1.24111e7i	− 1.48667i	0.668919	−	0.743335i	$-0.266757\pi$
$588$	− 2.38755e6i	− 0.284780i
$589$	−1.51473e7	−1.79906
$590$	0	0
$591$	−4.45592e6	−0.524770
$592$	− 1.47320e6i	− 0.172766i
$593$	− 8.56835e6i	− 1.00060i	−0.865852	−	0.500300i	$-0.833223\pi$
$593$	− 8.56835e6i	− 1.00060i	0.865852	−	0.500300i	$-0.166777\pi$
$594$	604930.	0.0703459
$595$	0	0
$596$	−4.87171e6	−0.561780
$597$	6.93583e6i	0.796457i
$598$	1.08669e7i	1.24267i
$599$	691285.	0.0787209	0.0393605	−	0.999225i	$-0.487468\pi$
$599$	691285.	0.0787209	0.0393605	+	0.999225i	$0.487468\pi$
$600$	0	0
$601$	−1.16271e7	−1.31306	−0.656528	−	0.754301i	$-0.727976\pi$
$601$	−1.16271e7	−1.31306	−0.656528	+	0.754301i	$0.727976\pi$
$602$	− 609797.i	− 0.0685794i
$603$	587712.i	0.0658220i
$604$	−2.40031e6	−0.267717
$605$	0	0
$606$	−1.91079e6	−0.211364
$607$	− 7.58227e6i	− 0.835271i	−0.908615	−	0.417635i	$-0.862859\pi$
$607$	− 7.58227e6i	− 0.835271i	0.908615	−	0.417635i	$-0.137141\pi$
$608$	2.10436e6i	0.230867i
$609$	742796.	0.0811570
$610$	0	0
$611$	3.70599e6	0.401606
$612$	32644.5i	0.00352316i
$613$	− 8.47440e6i	− 0.910873i	−0.890268	−	0.455436i	$-0.849483\pi$
$613$	− 8.47440e6i	− 0.910873i	0.890268	−	0.455436i	$-0.150517\pi$
$614$	7.32446e6	0.784070
$615$	0	0
$616$	199944.	0.0212303
$617$	− 6.17092e6i	− 0.652585i	−0.945269	−	0.326293i	$-0.894201\pi$
$617$	− 6.17092e6i	− 0.652585i	0.945269	−	0.326293i	$-0.105799\pi$
$618$	− 4.14721e6i	− 0.436802i
$619$	−1.32316e7	−1.38799	−0.693996	−	0.719979i	$-0.744151\pi$
$619$	−1.32316e7	−1.38799	−0.693996	+	0.719979i	$0.744151\pi$
$620$	0	0
$621$	−3.21934e6	−0.334995
$622$	7.92618e6i	0.821463i
$623$	1.51678e6i	0.156568i
$624$	−1.41739e6	−0.145723
$625$	0	0
$626$	1.04759e6	0.106845
$627$	3.83691e6i	0.389774i
$628$	− 323474.i	− 0.0327296i
$629$	144953.	0.0146084
$630$	0	0
$631$	8.36289e6	0.836148	0.418074	−	0.908413i	$-0.362705\pi$
$631$	8.36289e6	0.836148	0.418074	+	0.908413i	$0.362705\pi$
$632$	978816.i	0.0974784i
$633$	− 1.32688e6i	− 0.131620i
$634$	−2.14705e6	−0.212138
$635$	0	0
$636$	−3.95520e6	−0.387727
$637$	1.01999e7i	0.995976i
$638$	− 4.54772e6i	− 0.442326i
$639$	−378052.	−0.0366268
$640$	0	0
$641$	3.95498e6	0.380188	0.190094	−	0.981766i	$-0.439121\pi$
$641$	3.95498e6	0.380188	0.190094	+	0.981766i	$0.439121\pi$
$642$	3.37507e6i	0.323180i
$643$	− 1.52163e7i	− 1.45138i	−0.688023	−	0.725689i	$-0.741521\pi$
$643$	− 1.52163e7i	− 1.45138i	0.688023	−	0.725689i	$-0.258479\pi$
$644$	−1.06407e6	−0.101101
$645$	0	0
$646$	−207055.	−0.0195211
$647$	− 278563.i	− 0.0261615i	−0.999914	−	0.0130808i	$-0.995836\pi$
$647$	− 278563.i	− 0.0261615i	0.999914	−	0.0130808i	$-0.00416386\pi$
$648$	− 419904.i	− 0.0392837i
$649$	−5.94824e6	−0.554341
$650$	0	0
$651$	999003.	0.0923877
$652$	6.22317e6i	0.573314i
$653$	1.83671e6i	0.168561i	0.996442	+	0.0842807i	$0.0268592\pi$
$653$	1.83671e6i	0.168561i	−0.996442	+	0.0842807i	$0.973141\pi$
$654$	−6.92300e6	−0.632922
$655$	0	0
$656$	−1.39813e6	−0.126849
$657$	6.22043e6i	0.562221i
$658$	362883.i	0.0326740i
$659$	−9.41168e6	−0.844216	−0.422108	−	0.906546i	$-0.638710\pi$
$659$	−9.41168e6	−0.844216	−0.422108	+	0.906546i	$0.638710\pi$
$660$	0	0
$661$	9.11144e6	0.811117	0.405559	−	0.914069i	$-0.367077\pi$
$661$	9.11144e6	0.811117	0.405559	+	0.914069i	$0.367077\pi$
$662$	− 1.23668e7i	− 1.09676i
$663$	− 139462.i	− 0.0123217i
$664$	972640.	0.0856114
$665$	0	0
$666$	−1.86452e6	−0.162886
$667$	2.42023e7i	2.10640i
$668$	− 642544.i	− 0.0557137i
$669$	−7.18415e6	−0.620597
$670$	0	0
$671$	−4.65185e6	−0.398859
$672$	− 138788.i	− 0.0118558i
$673$	− 2.56281e6i	− 0.218112i	−0.994036	−	0.109056i	$-0.965217\pi$
$673$	− 2.56281e6i	− 0.218112i	0.994036	−	0.109056i	$-0.0347828\pi$
$674$	−1.60280e6	−0.135903
$675$	0	0
$676$	114600.	0.00964537
$677$	− 1.30035e7i	− 1.09041i	−0.838304	−	0.545203i	$-0.816453\pi$
$677$	− 1.30035e7i	− 1.09041i	0.838304	−	0.545203i	$-0.183547\pi$
$678$	− 168122.i	− 0.0140459i
$679$	1.88357e6	0.156786
$680$	0	0
$681$	1.04657e7	0.864769
$682$	− 6.11634e6i	− 0.503536i
$683$	1.22880e7i	1.00793i	0.863725	+	0.503963i	$0.168125\pi$
$683$	1.22880e7i	1.00793i	−0.863725	+	0.503963i	$0.831875\pi$
$684$	2.66334e6	0.217664
$685$	0	0
$686$	−2.01118e6	−0.163170
$687$	8.58530e6i	0.694007i
$688$	2.59152e6i	0.208729i
$689$	1.68971e7	1.35602
$690$	0	0
$691$	1.26029e7	1.00410	0.502050	−	0.864839i	$-0.332580\pi$
$691$	1.26029e7	1.00410	0.502050	+	0.864839i	$0.332580\pi$
$692$	− 3.50203e6i	− 0.278006i
$693$	− 253054.i	− 0.0200161i
$694$	4.21710e6	0.332365
$695$	0	0
$696$	−3.15674e6	−0.247011
$697$	− 137567.i	− 0.0107259i
$698$	1.21994e6i	0.0947760i
$699$	−5.65415e6	−0.437698
$700$	0	0
$701$	1.09248e7	0.839691	0.419846	−	0.907596i	$-0.362084\pi$
$701$	1.09248e7	0.839691	0.419846	+	0.907596i	$0.362084\pi$
$702$	1.79389e6i	0.137389i
$703$	− 1.18262e7i	− 0.902518i
$704$	−849724.	−0.0646169
$705$	0	0
$706$	1.22885e7	0.927869
$707$	799320.i	0.0601412i
$708$	4.12889e6i	0.309564i
$709$	2.12115e7	1.58473	0.792364	−	0.610048i	$-0.208850\pi$
$709$	2.12115e7	1.58473	0.792364	+	0.610048i	$0.208850\pi$
$710$	0	0
$711$	1.23881e6	0.0919035
$712$	− 6.44602e6i	− 0.476531i
$713$	3.25502e7i	2.39789i
$714$	13655.8	0.00100247
$715$	0	0
$716$	−5.57263e6	−0.406235
$717$	2.61011e6i	0.189610i
$718$	9.55040e6i	0.691370i
$719$	−1.13833e7	−0.821198	−0.410599	−	0.911816i	$-0.634680\pi$
$719$	−1.13833e7	−0.821198	−0.410599	+	0.911816i	$0.634680\pi$
$720$	0	0
$721$	−1.73486e6	−0.124287
$722$	6.98842e6i	0.498926i
$723$	− 1.59210e7i	− 1.13273i
$724$	3.95939e6	0.280725
$725$	0	0
$726$	4.24853e6	0.299155
$727$	2.37912e7i	1.66947i	0.550648	+	0.834737i	$0.314381\pi$
$727$	2.37912e7i	1.66947i	−0.550648	+	0.834737i	$0.685619\pi$
$728$	592922.i	0.0414638i
$729$	−531441.	−0.0370370
$730$	0	0
$731$	−254988.	−0.0176493
$732$	3.22902e6i	0.222737i
$733$	− 2.77719e6i	− 0.190918i	−0.995433	−	0.0954588i	$-0.969568\pi$
$733$	− 2.77719e6i	− 0.190918i	0.995433	−	0.0954588i	$-0.0304318\pi$
$734$	−6.22369e6	−0.426391
$735$	0	0
$736$	4.52209e6	0.307712
$737$	− 1.50521e6i	− 0.102077i
$738$	1.76951e6i	0.119595i
$739$	2.20856e7	1.48764	0.743821	−	0.668379i	$-0.233012\pi$
$739$	2.20856e7	1.48764	0.743821	+	0.668379i	$0.233012\pi$
$740$	0	0
$741$	−1.13781e7	−0.761247
$742$	1.65454e6i	0.110323i
$743$	− 576821.i	− 0.0383327i	−0.999816	−	0.0191663i	$-0.993899\pi$
$743$	− 576821.i	− 0.0383327i	0.999816	−	0.0191663i	$-0.00610121\pi$
$744$	−4.24557e6	−0.281193
$745$	0	0
$746$	779735.	0.0512979
$747$	− 1.23100e6i	− 0.0807152i
$748$	− 83607.1i	− 0.00546373i
$749$	1.41186e6	0.0919572
$750$	0	0
$751$	−1.09334e7	−0.707382	−0.353691	−	0.935362i	$-0.615074\pi$
$751$	−1.09334e7	−0.707382	−0.353691	+	0.935362i	$0.615074\pi$
$752$	− 1.54218e6i	− 0.0994470i
$753$	− 1.13636e7i	− 0.730347i
$754$	1.34860e7	0.863884
$755$	0	0
$756$	−175654.	−0.0111777
$757$	2.13095e6i	0.135155i	0.997714	+	0.0675777i	$0.0215270\pi$
$757$	2.13095e6i	0.135155i	−0.997714	+	0.0675777i	$0.978473\pi$
$758$	− 4.79748e6i	− 0.303278i
$759$	8.24517e6	0.519512
$760$	0	0
$761$	−4.25293e6	−0.266211	−0.133106	−	0.991102i	$-0.542495\pi$
$761$	−4.25293e6	−0.266211	−0.133106	+	0.991102i	$0.542495\pi$
$762$	− 1.89298e6i	− 0.118102i
$763$	2.89602e6i	0.180091i
$764$	−9.67194e6	−0.599487
$765$	0	0
$766$	1.24787e7	0.768417
$767$	− 1.76392e7i	− 1.08265i
$768$	589824.i	0.0360844i
$769$	−2.59288e7	−1.58113	−0.790564	−	0.612380i	$-0.790212\pi$
$769$	−2.59288e7	−1.58113	−0.790564	+	0.612380i	$0.790212\pi$
$770$	0	0
$771$	−8.30730e6	−0.503296
$772$	3.38600e6i	0.204477i
$773$	− 1.92730e7i	− 1.16012i	−0.814575	−	0.580058i	$-0.803030\pi$
$773$	− 1.92730e7i	− 1.16012i	0.814575	−	0.580058i	$-0.196970\pi$
$774$	3.27989e6	0.196792
$775$	0	0
$776$	−8.00482e6	−0.477196
$777$	779966.i	0.0463472i
$778$	− 1.47598e7i	− 0.874242i
$779$	−1.12235e7	−0.662652
$780$	0	0
$781$	968243.	0.0568011
$782$	444944.i	0.0260189i
$783$	3.99525e6i	0.232884i
$784$	4.24453e6	0.246627
$785$	0	0
$786$	1.03604e7	0.598164
$787$	− 1.08492e7i	− 0.624398i	−0.950017	−	0.312199i	$-0.898934\pi$
$787$	− 1.08492e7i	− 0.624398i	0.950017	−	0.312199i	$-0.101066\pi$
$788$	− 7.92164e6i	− 0.454464i
$789$	4.91408e6	0.281028
$790$	0	0
$791$	−70328.6	−0.00399660
$792$	1.07543e6i	0.0609214i
$793$	− 1.37948e7i	− 0.778990i
$794$	−9.84551e6	−0.554226
$795$	0	0
$796$	−1.23304e7	−0.689752
$797$	1.25477e7i	0.699712i	0.936804	+	0.349856i	$0.113769\pi$
$797$	1.25477e7i	0.699712i	−0.936804	+	0.349856i	$0.886231\pi$
$798$	− 1.11413e6i	− 0.0619337i
$799$	151741.	0.00840882
$800$	0	0
$801$	−8.15824e6	−0.449278
$802$	1.86882e7i	1.02596i
$803$	− 1.59314e7i	− 0.871896i
$804$	−1.04482e6	−0.0570035
$805$	0	0
$806$	1.81377e7	0.983430
$807$	5.33898e6i	0.288585i
$808$	− 3.39696e6i	− 0.183047i
$809$	2.10815e7	1.13248	0.566239	−	0.824241i	$-0.308398\pi$
$809$	2.10815e7	1.13248	0.566239	+	0.824241i	$0.308398\pi$
$810$	0	0
$811$	2.73952e7	1.46259	0.731295	−	0.682061i	$-0.238916\pi$
$811$	2.73952e7	1.46259	0.731295	+	0.682061i	$0.238916\pi$
$812$	1.32053e6i	0.0702840i
$813$	1.01916e6i	0.0540775i
$814$	4.77530e6	0.252604
$815$	0	0
$816$	−58034.7	−0.00305114
$817$	2.08035e7i	1.09039i
$818$	− 9.43259e6i	− 0.492887i
$819$	750417.	0.0390925
$820$	0	0
$821$	−3.26724e7	−1.69170	−0.845850	−	0.533421i	$-0.820906\pi$
$821$	−3.26724e7	−1.69170	−0.845850	+	0.533421i	$0.820906\pi$
$822$	1.16627e7i	0.602033i
$823$	1.70637e7i	0.878158i	0.898448	+	0.439079i	$0.144695\pi$
$823$	1.70637e7i	0.878158i	−0.898448	+	0.439079i	$0.855305\pi$
$824$	7.37282e6	0.378282
$825$	0	0
$826$	1.72719e6	0.0880828
$827$	− 1.79539e7i	− 0.912838i	−0.889765	−	0.456419i	$-0.849132\pi$
$827$	− 1.79539e7i	− 0.912838i	0.889765	−	0.456419i	$-0.150868\pi$
$828$	− 5.72328e6i	− 0.290114i
$829$	−3.29402e7	−1.66472	−0.832358	−	0.554239i	$-0.813010\pi$
$829$	−3.29402e7	−1.66472	−0.832358	+	0.554239i	$0.813010\pi$
$830$	0	0
$831$	−1.41101e7	−0.708805
$832$	− 2.51981e6i	− 0.126200i
$833$	417634.i	0.0208537i
$834$	−3.35575e6	−0.167061
$835$	0	0
$836$	−6.82117e6	−0.337554
$837$	5.37330e6i	0.265111i
$838$	2.10997e6i	0.103792i
$839$	3.83613e7	1.88143	0.940715	−	0.339198i	$-0.110156\pi$
$839$	3.83613e7	1.88143	0.940715	+	0.339198i	$0.110156\pi$
$840$	0	0
$841$	9.52422e6	0.464344
$842$	− 1.10190e7i	− 0.535624i
$843$	1.55108e7i	0.751734i
$844$	2.35889e6	0.113986
$845$	0	0
$846$	−1.95183e6	−0.0937595
$847$	− 1.77724e6i	− 0.0851212i
$848$	− 7.03147e6i	− 0.335781i
$849$	2.19673e7	1.04594
$850$	0	0
$851$	−2.54134e7	−1.20293
$852$	− 672093.i	− 0.0317198i
$853$	2.27088e7i	1.06861i	0.845290	+	0.534307i	$0.179427\pi$
$853$	2.27088e7i	1.06861i	−0.845290	+	0.534307i	$0.820573\pi$
$854$	1.35076e6	0.0633772
$855$	0	0
$856$	−6.00012e6	−0.279882
$857$	− 1.96555e7i	− 0.914183i	−0.889420	−	0.457091i	$-0.848891\pi$
$857$	− 1.96555e7i	− 0.914183i	0.889420	−	0.457091i	$-0.151109\pi$
$858$	− 4.59439e6i	− 0.213064i
$859$	7.98142e6	0.369060	0.184530	−	0.982827i	$-0.440924\pi$
$859$	7.98142e6	0.369060	0.184530	+	0.982827i	$0.440924\pi$
$860$	0	0
$861$	740221.	0.0340293
$862$	1.81763e7i	0.833178i
$863$	3.46241e7i	1.58253i	0.611474	+	0.791265i	$0.290577\pi$
$863$	3.46241e7i	1.58253i	−0.611474	+	0.791265i	$0.709423\pi$
$864$	746496.	0.0340207
$865$	0	0
$866$	4.72014e6	0.213875
$867$	1.27730e7i	0.577092i
$868$	1.77600e6i	0.0800101i
$869$	−3.17277e6	−0.142524
$870$	0	0
$871$	4.46361e6	0.199361
$872$	− 1.23076e7i	− 0.548126i
$873$	1.01311e7i	0.449905i
$874$	3.63012e7	1.60747
$875$	0	0
$876$	−1.10585e7	−0.486898
$877$	2.25448e7i	0.989802i	0.868949	+	0.494901i	$0.164796\pi$
$877$	2.25448e7i	0.989802i	−0.868949	+	0.494901i	$0.835204\pi$
$878$	1.45017e7i	0.634866i
$879$	1.97331e7	0.861435
$880$	0	0
$881$	1.89920e7	0.824388	0.412194	−	0.911096i	$-0.364763\pi$
$881$	1.89920e7	0.824388	0.412194	+	0.911096i	$0.364763\pi$
$882$	− 5.37199e6i	− 0.232522i
$883$	2.23335e7i	0.963952i	0.876185	+	0.481976i	$0.160081\pi$
$883$	2.23335e7i	0.963952i	−0.876185	+	0.481976i	$0.839919\pi$
$884$	247932.	0.0106709
$885$	0	0
$886$	−5.11487e6	−0.218902
$887$	− 2.11288e7i	− 0.901708i	−0.892598	−	0.450854i	$-0.851119\pi$
$887$	− 2.11288e7i	− 0.901708i	0.892598	−	0.450854i	$-0.148881\pi$
$888$	− 3.31471e6i	− 0.141063i
$889$	−791869.	−0.0336046
$890$	0	0
$891$	1.36109e6	0.0574372
$892$	− 1.27718e7i	− 0.537453i
$893$	− 1.23799e7i	− 0.519504i
$894$	−1.09614e7	−0.458691
$895$	0	0
$896$	246735.	0.0102674
$897$	2.44506e7i	1.01463i
$898$	− 1.30390e7i	− 0.539577i
$899$	4.03952e7	1.66698
$900$	0	0
$901$	691849.	0.0283922
$902$	− 4.53196e6i	− 0.185468i
$903$	− 1.37204e6i	− 0.0559949i
$904$	298883.	0.0121641
$905$	0	0
$906$	−5.40071e6	−0.218590
$907$	3.84724e6i	0.155286i	0.996981	+	0.0776428i	$0.0247394\pi$
$907$	3.84724e6i	0.155286i	−0.996981	+	0.0776428i	$0.975261\pi$
$908$	1.86057e7i	0.748912i
$909$	−4.29928e6	−0.172578
$910$	0	0
$911$	2.83893e7	1.13334	0.566668	−	0.823946i	$-0.308232\pi$
$911$	2.83893e7	1.13334	0.566668	+	0.823946i	$0.308232\pi$
$912$	4.73482e6i	0.188502i
$913$	3.15275e6i	0.125174i
$914$	−2.85861e7	−1.13185
$915$	0	0
$916$	−1.52628e7	−0.601027
$917$	− 4.33396e6i	− 0.170201i
$918$	73450.2i	0.00287665i
$919$	−6.64214e6	−0.259430	−0.129715	−	0.991551i	$-0.541406\pi$
$919$	−6.64214e6	−0.259430	−0.129715	+	0.991551i	$0.541406\pi$
$920$	0	0
$921$	1.64800e7	0.640190
$922$	− 1.58774e7i	− 0.615108i
$923$	2.87127e6i	0.110935i
$924$	449874.	0.0173345
$925$	0	0
$926$	−1.21105e7	−0.464125
$927$	− 9.33123e6i	− 0.356648i
$928$	− 5.61198e6i	− 0.213918i
$929$	−3.63569e7	−1.38213	−0.691063	−	0.722795i	$-0.742857\pi$
$929$	−3.63569e7	−1.38213	−0.691063	+	0.722795i	$0.742857\pi$
$930$	0	0
$931$	3.40731e7	1.28836
$932$	− 1.00518e7i	− 0.379058i
$933$	1.78339e7i	0.670722i
$934$	1.55481e7	0.583189
$935$	0	0
$936$	−3.18913e6	−0.118982
$937$	1.82221e7i	0.678032i	0.940781	+	0.339016i	$0.110094\pi$
$937$	1.82221e7i	0.678032i	−0.940781	+	0.339016i	$0.889906\pi$
$938$	437069.i	0.0162197i
$939$	2.35708e6	0.0872389
$940$	0	0
$941$	−2.65605e7	−0.977827	−0.488914	−	0.872332i	$-0.662607\pi$
$941$	−2.65605e7	−0.977827	−0.488914	+	0.872332i	$0.662607\pi$
$942$	− 727817.i	− 0.0267236i
$943$	2.41184e7i	0.883220i
$944$	−7.34025e6	−0.268090
$945$	0	0
$946$	−8.40025e6	−0.305186
$947$	− 1.85090e7i	− 0.670669i	−0.942099	−	0.335335i	$-0.891151\pi$
$947$	− 1.85090e7i	− 0.670669i	0.942099	−	0.335335i	$-0.108849\pi$
$948$	2.20234e6i	0.0795908i
$949$	4.72436e7	1.70285
$950$	0	0
$951$	−4.83086e6	−0.173210
$952$	24277.0i	0 0.000868167i
$953$	1.52191e7i	0.542820i	0.962464	+	0.271410i	$0.0874899\pi$
$953$	1.52191e7i	0.542820i	−0.962464	+	0.271410i	$0.912510\pi$
$954$	−8.89920e6	−0.316577
$955$	0	0
$956$	−4.64019e6	−0.164207
$957$	− 1.02324e7i	− 0.361158i
$958$	− 2.15535e7i	− 0.758759i
$959$	4.87874e6	0.171302
$960$	0	0
$961$	2.56993e7	0.897662
$962$	1.41609e7i	0.493347i
$963$	7.59390e6i	0.263876i
$964$	2.83041e7	0.980972
$965$	0	0
$966$	−2.39416e6	−0.0825486
$967$	− 3.25868e7i	− 1.12066i	−0.828268	−	0.560332i	$-0.810673\pi$
$967$	− 3.25868e7i	− 1.12066i	0.828268	−	0.560332i	$-0.189327\pi$
$968$	7.55294e6i	0.259076i
$969$	−465875.	−0.0159389
$970$	0	0
$971$	−4.92407e7	−1.67601	−0.838003	−	0.545665i	$-0.816277\pi$
$971$	−4.92407e7	−1.67601	−0.838003	+	0.545665i	$0.816277\pi$
$972$	− 944784.i	− 0.0320750i
$973$	1.40377e6i	0.0475352i
$974$	3.34880e7	1.13108
$975$	0	0
$976$	−5.74047e6	−0.192896
$977$	4.35216e7i	1.45871i	0.684137	+	0.729354i	$0.260179\pi$
$977$	4.35216e7i	1.45871i	−0.684137	+	0.729354i	$0.739821\pi$
$978$	1.40021e7i	0.468109i
$979$	2.08944e7	0.696743
$980$	0	0
$981$	−1.55767e7	−0.516778
$982$	− 1.38792e7i	− 0.459287i
$983$	1.76935e7i	0.584023i	0.956415	+	0.292012i	$0.0943246\pi$
$983$	1.76935e7i	0.584023i	−0.956415	+	0.292012i	$0.905675\pi$
$984$	−3.14580e6	−0.103572
$985$	0	0
$986$	552182.	0.0180880
$987$	816487.i	0.0266782i
$988$	− 2.02278e7i	− 0.659259i
$989$	4.47048e7	1.45333
$990$	0	0
$991$	−5.51638e7	−1.78431	−0.892154	−	0.451732i	$-0.850806\pi$
$991$	−5.51638e7	−1.78431	−0.892154	+	0.451732i	$0.850806\pi$
$992$	− 7.54768e6i	− 0.243520i
$993$	− 2.78254e7i	− 0.895505i
$994$	−281149.	−0.00902549
$995$	0	0
$996$	2.18844e6	0.0699014
$997$	− 4.84748e7i	− 1.54447i	−0.635340	−	0.772233i	$-0.719140\pi$
$997$	− 4.84748e7i	− 1.54447i	0.635340	−	0.772233i	$-0.280860\pi$
$998$	− 6.02163e6i	− 0.191376i
$999$	−4.19518e6	−0.132995

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	750.6.c.a.499.7		8
5.2	odd	4		750.6.a.d.1.2	✓	4
5.3	odd	4		750.6.a.e.1.3	yes	4
5.4	even	2	inner	750.6.c.a.499.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
750.6.a.d.1.2	✓	4	5.2	odd	4
750.6.a.e.1.3	yes	4	5.3	odd	4
750.6.c.a.499.2		8	5.4	even	2	inner
750.6.c.a.499.7		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	750.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} +15.0595i q^{7} -64.0000i q^{8} -81.0000 q^{9} +O(q^{10})\)	\(q+4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} +15.0595i q^{7} -64.0000i q^{8} -81.0000 q^{9} +207.452 q^{11} -144.000i q^{12} +615.187i q^{13} -60.2380 q^{14} +256.000 q^{16} +25.1887i q^{17} -324.000i q^{18} +2055.04 q^{19} -135.535 q^{21} +829.808i q^{22} -4416.11i q^{23} +576.000 q^{24} -2460.75 q^{26} -729.000i q^{27} -240.952i q^{28} -5480.45 q^{29} -7370.78 q^{31} +1024.00i q^{32} +1867.07i q^{33} -100.755 q^{34} +1296.00 q^{36} -5754.71i q^{37} +8220.17i q^{38} -5536.69 q^{39} -5461.45 q^{41} -542.142i q^{42} +10123.1i q^{43} -3319.23 q^{44} +17664.4 q^{46} -6024.16i q^{47} +2304.00i q^{48} +16580.2 q^{49} -226.698 q^{51} -9843.00i q^{52} -27466.7i q^{53} +2916.00 q^{54} +963.808 q^{56} +18495.4i q^{57} -21921.8i q^{58} -28672.9 q^{59} -22423.7 q^{61} -29483.1i q^{62} -1219.82i q^{63} -4096.00 q^{64} -7468.27 q^{66} -7255.70i q^{67} -403.019i q^{68} +39745.0 q^{69} +4667.31 q^{71} +5184.00i q^{72} -76795.5i q^{73} +23018.8 q^{74} -32880.7 q^{76} +3124.12i q^{77} -22146.7i q^{78} -15294.0 q^{79} +6561.00 q^{81} -21845.8i q^{82} +15197.5i q^{83} +2168.57 q^{84} -40492.5 q^{86} -49324.1i q^{87} -13276.9i q^{88} +100719. q^{89} -9264.41 q^{91} +70657.7i q^{92} -66337.1i q^{93} +24096.6 q^{94} -9216.00 q^{96} -125075. i q^{97} +66320.8i q^{98} -16803.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 128 q^{4} - 288 q^{6} - 648 q^{9}+O(q^{10}) \) 8 * q - 128 * q^4 - 288 * q^6 - 648 * q^9	\( 8 q - 128 q^{4} - 288 q^{6} - 648 q^{9} - 520 q^{11} + 688 q^{14} + 2048 q^{16} + 2672 q^{19} + 1548 q^{21} + 4608 q^{24} - 1216 q^{26} - 7072 q^{29} - 26636 q^{31} - 8592 q^{34} + 10368 q^{36} - 2736 q^{39} + 21172 q^{41} + 8320 q^{44} - 7680 q^{46} + 75868 q^{49} - 19332 q^{51} + 23328 q^{54} - 11008 q^{56} + 31180 q^{59} - 76620 q^{61} - 32768 q^{64} + 18720 q^{66} - 17280 q^{69} - 164820 q^{71} - 114944 q^{74} - 42752 q^{76} + 342456 q^{79} + 52488 q^{81} - 24768 q^{84} + 52432 q^{86} + 356856 q^{89} - 340324 q^{91} + 118720 q^{94} - 73728 q^{96} + 42120 q^{99}+O(q^{100}) \) 8 * q - 128 * q^4 - 288 * q^6 - 648 * q^9 - 520 * q^11 + 688 * q^14 + 2048 * q^16 + 2672 * q^19 + 1548 * q^21 + 4608 * q^24 - 1216 * q^26 - 7072 * q^29 - 26636 * q^31 - 8592 * q^34 + 10368 * q^36 - 2736 * q^39 + 21172 * q^41 + 8320 * q^44 - 7680 * q^46 + 75868 * q^49 - 19332 * q^51 + 23328 * q^54 - 11008 * q^56 + 31180 * q^59 - 76620 * q^61 - 32768 * q^64 + 18720 * q^66 - 17280 * q^69 - 164820 * q^71 - 114944 * q^74 - 42752 * q^76 + 342456 * q^79 + 52488 * q^81 - 24768 * q^84 + 52432 * q^86 + 356856 * q^89 - 340324 * q^91 + 118720 * q^94 - 73728 * q^96 + 42120 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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