LMFDB - Embedded newform 525.6.d.p.274.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,6,Mod(274,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("525.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	525.d (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:	$84.2015054018$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 322x^{10} + 38621x^{8} + 2116025x^{6} + 52188850x^{4} + 497085625x^{2} + 1556302500 $ x^12 + 322x^10 + 38621x^8 + 2116025x^6 + 52188850x^4 + 497085625*x^2 + 1556302500
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.9
Root			$6.12018i$ of defining polynomial
Character	$\chi$	$=$	525.274
Dual form			525.6.d.p.274.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.12018i q^{2} -9.00000i q^{3} -18.6970 q^{4} +64.0817 q^{6} -49.0000i q^{7} +94.7196i q^{8} -81.0000 q^{9} +O(q^{10})$	\(q+7.12018i q^{2} -9.00000i q^{3} -18.6970 q^{4} +64.0817 q^{6} -49.0000i q^{7} +94.7196i q^{8} -81.0000 q^{9} +643.995 q^{11} +168.273i q^{12} -483.073i q^{13} +348.889 q^{14} -1272.73 q^{16} -295.180i q^{17} -576.735i q^{18} +1449.70 q^{19} -441.000 q^{21} +4585.36i q^{22} -3927.79i q^{23} +852.476 q^{24} +3439.57 q^{26} +729.000i q^{27} +916.155i q^{28} -7472.33 q^{29} -860.228 q^{31} -6031.02i q^{32} -5795.95i q^{33} +2101.74 q^{34} +1514.46 q^{36} +404.951i q^{37} +10322.1i q^{38} -4347.65 q^{39} -11022.8 q^{41} -3140.00i q^{42} +10013.6i q^{43} -12040.8 q^{44} +27966.6 q^{46} +15471.1i q^{47} +11454.5i q^{48} -2401.00 q^{49} -2656.62 q^{51} +9032.02i q^{52} -34943.6i q^{53} -5190.61 q^{54} +4641.26 q^{56} -13047.3i q^{57} -53204.4i q^{58} -44925.9 q^{59} +38909.3 q^{61} -6124.98i q^{62} +3969.00i q^{63} +2214.73 q^{64} +41268.2 q^{66} -1348.37i q^{67} +5518.99i q^{68} -35350.1 q^{69} +31533.2 q^{71} -7672.29i q^{72} -10557.5i q^{73} -2883.33 q^{74} -27105.1 q^{76} -31555.7i q^{77} -30956.1i q^{78} -20049.3 q^{79} +6561.00 q^{81} -78484.1i q^{82} -83599.8i q^{83} +8245.39 q^{84} -71298.4 q^{86} +67251.0i q^{87} +60998.9i q^{88} +101941. q^{89} -23670.6 q^{91} +73438.0i q^{92} +7742.05i q^{93} -110157. q^{94} -54279.2 q^{96} -145504. i q^{97} -17095.6i q^{98} -52163.6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 264 q^{4} + 72 q^{6} - 972 q^{9}+O(q^{10}) $ 12 * q - 264 * q^4 + 72 * q^6 - 972 * q^9	$ 12 q - 264 q^{4} + 72 q^{6} - 972 q^{9} + 392 q^{14} + 3568 q^{16} - 520 q^{19} - 5292 q^{21} - 5454 q^{24} - 6534 q^{26} + 4244 q^{29} + 4524 q^{31} - 24778 q^{34} + 21384 q^{36} + 3816 q^{39} + 16248 q^{41} - 62322 q^{44} + 58720 q^{46} - 28812 q^{49} + 67320 q^{51} - 5832 q^{54} - 29694 q^{56} - 51316 q^{59} + 82376 q^{61} - 131894 q^{64} + 124218 q^{66} + 33336 q^{69} + 35160 q^{71} - 7942 q^{74} + 507092 q^{76} + 19472 q^{79} + 78732 q^{81} + 116424 q^{84} + 287632 q^{86} + 660720 q^{89} + 20776 q^{91} + 747360 q^{94} + 491220 q^{96}+O(q^{100}) $ 12 * q - 264 * q^4 + 72 * q^6 - 972 * q^9 + 392 * q^14 + 3568 * q^16 - 520 * q^19 - 5292 * q^21 - 5454 * q^24 - 6534 * q^26 + 4244 * q^29 + 4524 * q^31 - 24778 * q^34 + 21384 * q^36 + 3816 * q^39 + 16248 * q^41 - 62322 * q^44 + 58720 * q^46 - 28812 * q^49 + 67320 * q^51 - 5832 * q^54 - 29694 * q^56 - 51316 * q^59 + 82376 * q^61 - 131894 * q^64 + 124218 * q^66 + 33336 * q^69 + 35160 * q^71 - 7942 * q^74 + 507092 * q^76 + 19472 * q^79 + 78732 * q^81 + 116424 * q^84 + 287632 * q^86 + 660720 * q^89 + 20776 * q^91 + 747360 * q^94 + 491220 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	7.12018i	1.25868i	0.777129	+	0.629341i	$0.216675\pi$
$2$	7.12018i	1.25868i	−0.777129	+	0.629341i	$0.783325\pi$
$3$	− 9.00000i	− 0.577350i
$3$	− 9.00000i	− 0.577350i
$4$	−18.6970	−0.584282
$5$	0	0
$5$	0	0
$6$	64.0817	0.726701
$7$	− 49.0000i	− 0.377964i
$7$	− 49.0000i	− 0.377964i
$8$	94.7196i	0.523257i
$9$	−81.0000	−0.333333
$10$	0	0
$11$	643.995	1.60472	0.802362	−	0.596837i	$-0.203576\pi$
$11$	643.995	1.60472	0.802362	+	0.596837i	$0.203576\pi$
$12$	168.273i	0.337336i
$13$	− 483.073i	− 0.792782i	−0.918082	−	0.396391i	$-0.870262\pi$
$13$	− 483.073i	− 0.792782i	0.918082	−	0.396391i	$-0.129738\pi$
$14$	348.889	0.475737
$15$	0	0
$16$	−1272.73	−1.24290
$17$	− 295.180i	− 0.247722i	−0.992300	−	0.123861i	$-0.960472\pi$
$17$	− 295.180i	− 0.247722i	0.992300	−	0.123861i	$-0.0395277\pi$
$18$	− 576.735i	− 0.419561i
$19$	1449.70	0.921285	0.460642	−	0.887586i	$-0.347619\pi$
$19$	1449.70	0.921285	0.460642	+	0.887586i	$0.347619\pi$
$20$	0	0
$21$	−441.000	−0.218218
$22$	4585.36i	2.01984i
$23$	− 3927.79i	− 1.54821i	−0.633060	−	0.774103i	$-0.718201\pi$
$23$	− 3927.79i	− 1.54821i	0.633060	−	0.774103i	$-0.281799\pi$
$24$	852.476	0.302102
$25$	0	0
$26$	3439.57	0.997861
$27$	729.000i	0.192450i
$28$	916.155i	0.220838i
$29$	−7472.33	−1.64991	−0.824956	−	0.565196i	$-0.808800\pi$
$29$	−7472.33	−1.64991	−0.824956	+	0.565196i	$0.808800\pi$
$30$	0	0
$31$	−860.228	−0.160772	−0.0803858	−	0.996764i	$-0.525615\pi$
$31$	−860.228	−0.160772	−0.0803858	+	0.996764i	$0.525615\pi$
$32$	− 6031.02i	− 1.04116i
$33$	− 5795.95i	− 0.926488i
$34$	2101.74	0.311804
$35$	0	0
$36$	1514.46	0.194761
$37$	404.951i	0.0486293i	0.999704	+	0.0243147i	$0.00774036\pi$
$37$	404.951i	0.0486293i	−0.999704	+	0.0243147i	$0.992260\pi$
$38$	10322.1i	1.15961i
$39$	−4347.65	−0.457713
$40$	0	0
$41$	−11022.8	−1.02407	−0.512036	−	0.858964i	$-0.671109\pi$
$41$	−11022.8	−1.02407	−0.512036	+	0.858964i	$0.671109\pi$
$42$	− 3140.00i	− 0.274667i
$43$	10013.6i	0.825881i	0.910758	+	0.412941i	$0.135498\pi$
$43$	10013.6i	0.825881i	−0.910758	+	0.412941i	$0.864502\pi$
$44$	−12040.8	−0.937612
$45$	0	0
$46$	27966.6	1.94870
$47$	15471.1i	1.02159i	0.859702	+	0.510796i	$0.170649\pi$
$47$	15471.1i	1.02159i	−0.859702	+	0.510796i	$0.829351\pi$
$48$	11454.5i	0.717587i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	−2656.62	−0.143022
$52$	9032.02i	0.463209i
$53$	− 34943.6i	− 1.70875i	−0.519661	−	0.854373i	$-0.673942\pi$
$53$	− 34943.6i	− 1.70875i	0.519661	−	0.854373i	$-0.326058\pi$
$54$	−5190.61	−0.242234
$55$	0	0
$56$	4641.26	0.197772
$57$	− 13047.3i	− 0.531904i
$58$	− 53204.4i	− 2.07672i
$59$	−44925.9	−1.68022	−0.840111	−	0.542414i	$-0.817510\pi$
$59$	−44925.9	−1.68022	−0.840111	+	0.542414i	$0.817510\pi$
$60$	0	0
$61$	38909.3	1.33884	0.669420	−	0.742884i	$-0.266543\pi$
$61$	38909.3	1.33884	0.669420	+	0.742884i	$0.266543\pi$
$62$	− 6124.98i	− 0.202360i
$63$	3969.00i	0.125988i
$64$	2214.73	0.0675882
$65$	0	0
$66$	41268.2	1.16615
$67$	− 1348.37i	− 0.0366963i	−0.999832	−	0.0183481i	$-0.994159\pi$
$67$	− 1348.37i	− 0.0366963i	0.999832	−	0.0183481i	$-0.00584072\pi$
$68$	5518.99i	0.144740i
$69$	−35350.1	−0.893857
$70$	0	0
$71$	31533.2	0.742373	0.371187	−	0.928558i	$-0.378951\pi$
$71$	31533.2	0.742373	0.371187	+	0.928558i	$0.378951\pi$
$72$	− 7672.29i	− 0.174419i
$73$	− 10557.5i	− 0.231875i	−0.993256	−	0.115938i	$-0.963013\pi$
$73$	− 10557.5i	− 0.231875i	0.993256	−	0.115938i	$-0.0369873\pi$
$74$	−2883.33	−0.0612089
$75$	0	0
$76$	−27105.1	−0.538290
$77$	− 31555.7i	− 0.606529i
$78$	− 30956.1i	− 0.576116i
$79$	−20049.3	−0.361436	−0.180718	−	0.983535i	$-0.557842\pi$
$79$	−20049.3	−0.361436	−0.180718	+	0.983535i	$0.557842\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	− 78484.1i	− 1.28898i
$83$	− 83599.8i	− 1.33202i	−0.745944	−	0.666009i	$-0.768001\pi$
$83$	− 83599.8i	− 1.33202i	0.745944	−	0.666009i	$-0.231999\pi$
$84$	8245.39	0.127501
$85$	0	0
$86$	−71298.4	−1.03952
$87$	67251.0i	0.952578i
$88$	60998.9i	0.839683i
$89$	101941.	1.36419	0.682093	−	0.731265i	$-0.261070\pi$
$89$	101941.	1.36419	0.682093	+	0.731265i	$0.261070\pi$
$90$	0	0
$91$	−23670.6	−0.299644
$92$	73438.0i	0.904589i
$93$	7742.05i	0.0928215i
$94$	−110157.	−1.28586
$95$	0	0
$96$	−54279.2	−0.601112
$97$	− 145504.i	− 1.57017i	−0.619389	−	0.785084i	$-0.712620\pi$
$97$	− 145504.i	− 1.57017i	0.619389	−	0.785084i	$-0.287380\pi$
$98$	− 17095.6i	− 0.179812i
$99$	−52163.6	−0.534908
$100$	0	0
$101$	107659.	1.05014	0.525072	−	0.851058i	$-0.324039\pi$
$101$	107659.	1.05014	0.525072	+	0.851058i	$0.324039\pi$
$102$	− 18915.6i	− 0.180020i
$103$	− 97762.4i	− 0.907986i	−0.891005	−	0.453993i	$-0.849999\pi$
$103$	− 97762.4i	− 0.907986i	0.891005	−	0.453993i	$-0.150001\pi$
$104$	45756.4	0.414829
$105$	0	0
$106$	248805.	2.15077
$107$	− 198515.i	− 1.67623i	−0.545495	−	0.838114i	$-0.683658\pi$
$107$	− 198515.i	− 1.67623i	0.545495	−	0.838114i	$-0.316342\pi$
$108$	− 13630.1i	− 0.112445i
$109$	26060.0	0.210091	0.105046	−	0.994467i	$-0.466501\pi$
$109$	26060.0	0.210091	0.105046	+	0.994467i	$0.466501\pi$
$110$	0	0
$111$	3644.56	0.0280761
$112$	62363.6i	0.469771i
$113$	− 133467.i	− 0.983280i	−0.870799	−	0.491640i	$-0.836398\pi$
$113$	− 133467.i	− 0.983280i	0.870799	−	0.491640i	$-0.163602\pi$
$114$	92899.1	0.669498
$115$	0	0
$116$	139710.	0.964015
$117$	39128.9i	0.264261i
$118$	− 319881.i	− 2.11487i
$119$	−14463.8	−0.0936302
$120$	0	0
$121$	253678.	1.57514
$122$	277041.i	1.68518i
$123$	99204.8i	0.591249i
$124$	16083.7	0.0939360
$125$	0	0
$126$	−28260.0	−0.158579
$127$	− 159246.i	− 0.876113i	−0.898948	−	0.438056i	$-0.855667\pi$
$127$	− 159246.i	− 0.876113i	0.898948	−	0.438056i	$-0.144333\pi$
$128$	− 177223.i	− 0.956084i
$129$	90122.1	0.476823
$130$	0	0
$131$	−297360.	−1.51393	−0.756963	−	0.653458i	$-0.773318\pi$
$131$	−297360.	−1.51393	−0.756963	+	0.653458i	$0.773318\pi$
$132$	108367.i	0.541331i
$133$	− 71035.3i	− 0.348213i
$134$	9600.64	0.0461890
$135$	0	0
$136$	27959.3	0.129622
$137$	− 127351.i	− 0.579698i	−0.957072	−	0.289849i	$-0.906395\pi$
$137$	− 127351.i	− 0.579698i	0.957072	−	0.289849i	$-0.0936052\pi$
$138$	− 251699.i	− 1.12508i
$139$	360420.	1.58224	0.791118	−	0.611663i	$-0.209499\pi$
$139$	360420.	1.58224	0.791118	+	0.611663i	$0.209499\pi$
$140$	0	0
$141$	139240.	0.589816
$142$	224522.i	0.934412i
$143$	− 311096.i	− 1.27220i
$144$	103091.	0.414299
$145$	0	0
$146$	75171.4	0.291857
$147$	21609.0i	0.0824786i
$148$	− 7571.38i	− 0.0284132i
$149$	103509.	0.381954	0.190977	−	0.981594i	$-0.438834\pi$
$149$	103509.	0.381954	0.190977	+	0.981594i	$0.438834\pi$
$150$	0	0
$151$	−313442.	−1.11870	−0.559352	−	0.828930i	$-0.688950\pi$
$151$	−313442.	−1.11870	−0.559352	+	0.828930i	$0.688950\pi$
$152$	137315.i	0.482068i
$153$	23909.6i	0.0825740i
$154$	224683.	0.763427
$155$	0	0
$156$	81288.2	0.267434
$157$	− 76981.7i	− 0.249252i	−0.992204	−	0.124626i	$-0.960227\pi$
$157$	− 76981.7i	− 0.249252i	0.992204	−	0.124626i	$-0.0397731\pi$
$158$	− 142755.i	− 0.454933i
$159$	−314492.	−0.986545
$160$	0	0
$161$	−192462.	−0.585167
$162$	46715.5i	0.139854i
$163$	319326.i	0.941381i	0.882298	+	0.470690i	$0.155995\pi$
$163$	319326.i	0.941381i	−0.882298	+	0.470690i	$0.844005\pi$
$164$	206093.	0.598347
$165$	0	0
$166$	595246.	1.67659
$167$	578553.i	1.60528i	0.596461	+	0.802642i	$0.296573\pi$
$167$	578553.i	1.60528i	−0.596461	+	0.802642i	$0.703427\pi$
$168$	− 41771.3i	− 0.114184i
$169$	137934.	0.371496
$170$	0	0
$171$	−117426.	−0.307095
$172$	− 187224.i	− 0.482548i
$173$	333874.i	0.848140i	0.905630	+	0.424070i	$0.139399\pi$
$173$	333874.i	0.848140i	−0.905630	+	0.424070i	$0.860601\pi$
$174$	−478839.	−1.19899
$175$	0	0
$176$	−819629.	−1.99451
$177$	404333.i	0.970077i
$178$	725839.i	1.71708i
$179$	−120870.	−0.281959	−0.140979	−	0.990013i	$-0.545025\pi$
$179$	−120870.	−0.281959	−0.140979	+	0.990013i	$0.545025\pi$
$180$	0	0
$181$	−349775.	−0.793583	−0.396792	−	0.917909i	$-0.629876\pi$
$181$	−349775.	−0.793583	−0.396792	+	0.917909i	$0.629876\pi$
$182$	− 168539.i	− 0.377156i
$183$	− 350184.i	− 0.772980i
$184$	372039.	0.810109
$185$	0	0
$186$	−55124.8	−0.116833
$187$	− 190094.i	− 0.397526i
$188$	− 289264.i	− 0.596898i
$189$	35721.0	0.0727393
$190$	0	0
$191$	427748.	0.848409	0.424204	−	0.905567i	$-0.360554\pi$
$191$	427748.	0.848409	0.424204	+	0.905567i	$0.360554\pi$
$192$	− 19932.6i	− 0.0390221i
$193$	− 995110.i	− 1.92299i	−0.274816	−	0.961497i	$-0.588617\pi$
$193$	− 995110.i	− 1.92299i	0.274816	−	0.961497i	$-0.411383\pi$
$194$	1.03602e6	1.97634
$195$	0	0
$196$	44891.6	0.0834689
$197$	311407.i	0.571693i	0.958275	+	0.285847i	$0.0922748\pi$
$197$	311407.i	0.571693i	−0.958275	+	0.285847i	$0.907725\pi$
$198$	− 371414.i	− 0.673280i
$199$	210791.	0.377329	0.188664	−	0.982042i	$-0.439584\pi$
$199$	210791.	0.377329	0.188664	+	0.982042i	$0.439584\pi$
$200$	0	0
$201$	−12135.3	−0.0211866
$202$	766555.i	1.32180i
$203$	366144.i	0.623608i
$204$	49670.9	0.0835655
$205$	0	0
$206$	696087.	1.14287
$207$	318151.i	0.516069i
$208$	614819.i	0.985346i
$209$	933598.	1.47841
$210$	0	0
$211$	817503.	1.26410	0.632052	−	0.774926i	$-0.282213\pi$
$211$	817503.	1.26410	0.632052	+	0.774926i	$0.282213\pi$
$212$	653341.i	0.998390i
$213$	− 283799.i	− 0.428609i
$214$	1.41346e6	2.10984
$215$	0	0
$216$	−69050.6	−0.100701
$217$	42151.2i	0.0607659i
$218$	185552.i	0.264438i
$219$	−95017.6	−0.133873
$220$	0	0
$221$	−142593.	−0.196390
$222$	25949.9i	0.0353390i
$223$	500897.i	0.674507i	0.941414	+	0.337253i	$0.109498\pi$
$223$	500897.i	0.674507i	−0.941414	+	0.337253i	$0.890502\pi$
$224$	−295520.	−0.393520
$225$	0	0
$226$	950308.	1.23764
$227$	308220.i	0.397006i	0.980100	+	0.198503i	$0.0636079\pi$
$227$	308220.i	0.397006i	−0.980100	+	0.198503i	$0.936392\pi$
$228$	243946.i	0.310782i
$229$	506195.	0.637865	0.318933	−	0.947777i	$-0.396676\pi$
$229$	506195.	0.637865	0.318933	+	0.947777i	$0.396676\pi$
$230$	0	0
$231$	−284002.	−0.350180
$232$	− 707776.i	− 0.863328i
$233$	− 325167.i	− 0.392388i	−0.980565	−	0.196194i	$-0.937142\pi$
$233$	− 325167.i	− 0.392388i	0.980565	−	0.196194i	$-0.0628583\pi$
$234$	−278605.	−0.332620
$235$	0	0
$236$	839981.	0.981724
$237$	180444.i	0.208675i
$238$	− 102985.i	− 0.117851i
$239$	−86838.9	−0.0983376	−0.0491688	−	0.998790i	$-0.515657\pi$
$239$	−86838.9	−0.0983376	−0.0491688	+	0.998790i	$0.515657\pi$
$240$	0	0
$241$	−597851.	−0.663056	−0.331528	−	0.943445i	$-0.607564\pi$
$241$	−597851.	−0.663056	−0.331528	+	0.943445i	$0.607564\pi$
$242$	1.80623e6i	1.98260i
$243$	− 59049.0i	− 0.0641500i
$244$	−727488.	−0.782261
$245$	0	0
$246$	−706357.	−0.744194
$247$	− 700310.i	− 0.730378i
$248$	− 81480.4i	− 0.0841248i
$249$	−752398.	−0.769041
$250$	0	0
$251$	−587176.	−0.588280	−0.294140	−	0.955762i	$-0.595033\pi$
$251$	−587176.	−0.588280	−0.294140	+	0.955762i	$0.595033\pi$
$252$	− 74208.5i	− 0.0736126i
$253$	− 2.52948e6i	− 2.48444i
$254$	1.13386e6	1.10275
$255$	0	0
$256$	1.33273e6	1.27099
$257$	1.06356e6i	1.00445i	0.864737	+	0.502225i	$0.167485\pi$
$257$	1.06356e6i	1.00445i	−0.864737	+	0.502225i	$0.832515\pi$
$258$	641686.i	0.600169i
$259$	19842.6	0.0183802
$260$	0	0
$261$	605259.	0.549971
$262$	− 2.11726e6i	− 1.90555i
$263$	60359.2i	0.0538089i	0.999638	+	0.0269045i	$0.00856499\pi$
$263$	60359.2i	0.0538089i	−0.999638	+	0.0269045i	$0.991435\pi$
$264$	548990.	0.484791
$265$	0	0
$266$	505784.	0.438290
$267$	− 917469.i	− 0.787613i
$268$	25210.5i	0.0214410i
$269$	−1.17737e6	−0.992044	−0.496022	−	0.868310i	$-0.665207\pi$
$269$	−1.17737e6	−0.992044	−0.496022	+	0.868310i	$0.665207\pi$
$270$	0	0
$271$	−603107.	−0.498851	−0.249426	−	0.968394i	$-0.580242\pi$
$271$	−603107.	−0.498851	−0.249426	+	0.968394i	$0.580242\pi$
$272$	375683.i	0.307893i
$273$	213035.i	0.172999i
$274$	906765.	0.729656
$275$	0	0
$276$	660942.	0.522265
$277$	− 2.07182e6i	− 1.62238i	−0.584785	−	0.811188i	$-0.698821\pi$
$277$	− 2.07182e6i	− 1.62238i	0.584785	−	0.811188i	$-0.301179\pi$
$278$	2.56625e6i	1.99153i
$279$	69678.4	0.0535905
$280$	0	0
$281$	−1.68039e6	−1.26953	−0.634766	−	0.772705i	$-0.718903\pi$
$281$	−1.68039e6	−1.26953	−0.634766	+	0.772705i	$0.718903\pi$
$282$	991416.i	0.742392i
$283$	1.22457e6i	0.908906i	0.890771	+	0.454453i	$0.150165\pi$
$283$	1.22457e6i	0.908906i	−0.890771	+	0.454453i	$0.849835\pi$
$284$	−589577.	−0.433755
$285$	0	0
$286$	2.21506e6	1.60129
$287$	540115.i	0.387063i
$288$	488512.i	0.347052i
$289$	1.33273e6	0.938634
$290$	0	0
$291$	−1.30954e6	−0.906537
$292$	197394.i	0.135481i
$293$	− 1.73030e6i	− 1.17748i	−0.808324	−	0.588738i	$-0.799625\pi$
$293$	− 1.73030e6i	− 1.17748i	0.808324	−	0.588738i	$-0.200375\pi$
$294$	−153860.	−0.103814
$295$	0	0
$296$	−38356.8	−0.0254456
$297$	469472.i	0.308829i
$298$	737002.i	0.480759i
$299$	−1.89741e6	−1.22739
$300$	0	0
$301$	490665.	0.312154
$302$	− 2.23177e6i	− 1.40809i
$303$	− 968934.i	− 0.606300i
$304$	−1.84507e6	−1.14506
$305$	0	0
$306$	−170241.	−0.103935
$307$	− 648387.i	− 0.392634i	−0.980540	−	0.196317i	$-0.937102\pi$
$307$	− 648387.i	− 0.392634i	0.980540	−	0.196317i	$-0.0628982\pi$
$308$	589999.i	0.354384i
$309$	−879862.	−0.524226
$310$	0	0
$311$	−643151.	−0.377061	−0.188531	−	0.982067i	$-0.560372\pi$
$311$	−643151.	−0.377061	−0.188531	+	0.982067i	$0.560372\pi$
$312$	− 411808.i	− 0.239501i
$313$	1.34024e6i	0.773255i	0.922236	+	0.386628i	$0.126360\pi$
$313$	1.34024e6i	0.773255i	−0.922236	+	0.386628i	$0.873640\pi$
$314$	548124.	0.313729
$315$	0	0
$316$	374862.	0.211181
$317$	− 202260.i	− 0.113048i	−0.998401	−	0.0565238i	$-0.981998\pi$
$317$	− 202260.i	− 0.113048i	0.998401	−	0.0565238i	$-0.0180017\pi$
$318$	− 2.23924e6i	− 1.24175i
$319$	−4.81214e6	−2.64766
$320$	0	0
$321$	−1.78663e6	−0.967771
$322$	− 1.37036e6i	− 0.736539i
$323$	− 427922.i	− 0.228223i
$324$	−122671.	−0.0649203
$325$	0	0
$326$	−2.27366e6	−1.18490
$327$	− 234540.i	− 0.121296i
$328$	− 1.04407e6i	− 0.535853i
$329$	758086.	0.386125
$330$	0	0
$331$	−171257.	−0.0859167	−0.0429583	−	0.999077i	$-0.513678\pi$
$331$	−171257.	−0.0859167	−0.0429583	+	0.999077i	$0.513678\pi$
$332$	1.56307e6i	0.778274i
$333$	− 32801.0i	− 0.0162098i
$334$	−4.11940e6	−2.02054
$335$	0	0
$336$	561272.	0.271222
$337$	− 1.13448e6i	− 0.544157i	−0.962275	−	0.272078i	$-0.912289\pi$
$337$	− 1.13448e6i	− 0.544157i	0.962275	−	0.272078i	$-0.0877110\pi$
$338$	982115.i	0.467596i
$339$	−1.20120e6	−0.567697
$340$	0	0
$341$	−553982.	−0.257994
$342$	− 836092.i	− 0.386535i
$343$	117649.i	0.0539949i
$344$	−948481.	−0.432148
$345$	0	0
$346$	−2.37724e6	−1.06754
$347$	536124.i	0.239024i	0.992833	+	0.119512i	$0.0381330\pi$
$347$	536124.i	0.239024i	−0.992833	+	0.119512i	$0.961867\pi$
$348$	− 1.25739e6i	− 0.556574i
$349$	2.69134e6	1.18278	0.591391	−	0.806385i	$-0.298579\pi$
$349$	2.69134e6	1.18278	0.591391	+	0.806385i	$0.298579\pi$
$350$	0	0
$351$	352160.	0.152571
$352$	− 3.88394e6i	− 1.67077i
$353$	2.16964e6i	0.926726i	0.886169	+	0.463363i	$0.153357\pi$
$353$	2.16964e6i	0.926726i	−0.886169	+	0.463363i	$0.846643\pi$
$354$	−2.87893e6	−1.22102
$355$	0	0
$356$	−1.90599e6	−0.797070
$357$	130174.i	0.0540574i
$358$	− 860616.i	− 0.354897i
$359$	3.42183e6	1.40127	0.700635	−	0.713519i	$-0.252900\pi$
$359$	3.42183e6	1.40127	0.700635	+	0.713519i	$0.252900\pi$
$360$	0	0
$361$	−374471.	−0.151234
$362$	− 2.49046e6i	− 0.998869i
$363$	− 2.28310e6i	− 0.909408i
$364$	442569.	0.175076
$365$	0	0
$366$	2.49337e6	0.972936
$367$	− 1.66671e6i	− 0.645945i	−0.946408	−	0.322972i	$-0.895318\pi$
$367$	− 1.66671e6i	− 0.645945i	0.946408	−	0.322972i	$-0.104682\pi$
$368$	4.99900e6i	1.92426i
$369$	892844.	0.341358
$370$	0	0
$371$	−1.71223e6	−0.645845
$372$	− 144753.i	− 0.0542340i
$373$	− 3.41660e6i	− 1.27152i	−0.771888	−	0.635759i	$-0.780687\pi$
$373$	− 3.41660e6i	− 1.27152i	0.771888	−	0.635759i	$-0.219313\pi$
$374$	1.35351e6	0.500359
$375$	0	0
$376$	−1.46542e6	−0.534555
$377$	3.60968e6i	1.30802i
$378$	254340.i	0.0915557i
$379$	424122.	0.151668	0.0758338	−	0.997120i	$-0.475838\pi$
$379$	424122.	0.151668	0.0758338	+	0.997120i	$0.475838\pi$
$380$	0	0
$381$	−1.43322e6	−0.505824
$382$	3.04565e6i	1.06788i
$383$	− 2.46043e6i	− 0.857065i	−0.903526	−	0.428532i	$-0.859031\pi$
$383$	− 2.46043e6i	− 0.857065i	0.903526	−	0.428532i	$-0.140969\pi$
$384$	−1.59501e6	−0.551995
$385$	0	0
$386$	7.08537e6	2.42044
$387$	− 811099.i	− 0.275294i
$388$	2.72050e6i	0.917422i
$389$	−2.04949e6	−0.686708	−0.343354	−	0.939206i	$-0.611563\pi$
$389$	−2.04949e6	−0.686708	−0.343354	+	0.939206i	$0.611563\pi$
$390$	0	0
$391$	−1.15941e6	−0.383525
$392$	− 227422.i	− 0.0747510i
$393$	2.67624e6i	0.874065i
$394$	−2.21728e6	−0.719581
$395$	0	0
$396$	975304.	0.312537
$397$	− 1.88417e6i	− 0.599990i	−0.953941	−	0.299995i	$-0.903015\pi$
$397$	− 1.88417e6i	− 0.599990i	0.953941	−	0.299995i	$-0.0969850\pi$
$398$	1.50087e6i	0.474937i
$399$	−639317.	−0.201041
$400$	0	0
$401$	−2.72204e6	−0.845344	−0.422672	−	0.906283i	$-0.638908\pi$
$401$	−2.72204e6	−0.845344	−0.422672	+	0.906283i	$0.638908\pi$
$402$	− 86405.8i	− 0.0266672i
$403$	415552.i	0.127457i
$404$	−2.01291e6	−0.613580
$405$	0	0
$406$	−2.60701e6	−0.784925
$407$	260786.i	0.0780367i
$408$	− 251634.i	− 0.0748375i
$409$	−6.71271e6	−1.98422	−0.992110	−	0.125369i	$-0.959988\pi$
$409$	−6.71271e6	−1.98422	−0.992110	+	0.125369i	$0.959988\pi$
$410$	0	0
$411$	−1.14616e6	−0.334689
$412$	1.82787e6i	0.530520i
$413$	2.20137e6i	0.635064i
$414$	−2.26529e6	−0.649567
$415$	0	0
$416$	−2.91342e6	−0.825410
$417$	− 3.24378e6i	− 0.913505i
$418$	6.64739e6i	1.86085i
$419$	−2.81730e6	−0.783966	−0.391983	−	0.919972i	$-0.628211\pi$
$419$	−2.81730e6	−0.783966	−0.391983	+	0.919972i	$0.628211\pi$
$420$	0	0
$421$	5.56287e6	1.52966	0.764828	−	0.644235i	$-0.222824\pi$
$421$	5.56287e6	1.52966	0.764828	+	0.644235i	$0.222824\pi$
$422$	5.82077e6i	1.59111i
$423$	− 1.25316e6i	− 0.340531i
$424$	3.30984e6	0.894113
$425$	0	0
$426$	2.02070e6	0.539483
$427$	− 1.90656e6i	− 0.506034i
$428$	3.71164e6i	0.979391i
$429$	−2.79986e6	−0.734503
$430$	0	0
$431$	1.36693e6	0.354448	0.177224	−	0.984171i	$-0.443288\pi$
$431$	1.36693e6	0.354448	0.177224	+	0.984171i	$0.443288\pi$
$432$	− 927817.i	− 0.239196i
$433$	− 1.51018e6i	− 0.387086i	−0.981092	−	0.193543i	$-0.938002\pi$
$433$	− 1.51018e6i	− 0.387086i	0.981092	−	0.193543i	$-0.0619979\pi$
$434$	−300124.	−0.0764850
$435$	0	0
$436$	−487244.	−0.122752
$437$	− 5.69411e6i	− 1.42634i
$438$	− 676543.i	− 0.168504i
$439$	2.67644e6	0.662821	0.331411	−	0.943487i	$-0.392475\pi$
$439$	2.67644e6	0.662821	0.331411	+	0.943487i	$0.392475\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	− 1.01529e6i	− 0.247192i
$443$	− 7.00758e6i	− 1.69652i	−0.529580	−	0.848260i	$-0.677651\pi$
$443$	− 7.00758e6i	− 1.69652i	0.529580	−	0.848260i	$-0.322349\pi$
$444$	−68142.4	−0.0164044
$445$	0	0
$446$	−3.56648e6	−0.848990
$447$	− 931579.i	− 0.220521i
$448$	− 108522.i	− 0.0255459i
$449$	1.08547e6	0.254098	0.127049	−	0.991896i	$-0.459449\pi$
$449$	1.08547e6	0.254098	0.127049	+	0.991896i	$0.459449\pi$
$450$	0	0
$451$	−7.09860e6	−1.64335
$452$	2.49543e6i	0.574513i
$453$	2.82098e6i	0.645884i
$454$	−2.19459e6	−0.499704
$455$	0	0
$456$	1.23583e6	0.278322
$457$	7.45190e6i	1.66908i	0.550950	+	0.834538i	$0.314266\pi$
$457$	7.45190e6i	1.66908i	−0.550950	+	0.834538i	$0.685734\pi$
$458$	3.60420e6i	0.802870i
$459$	215186.	0.0476741
$460$	0	0
$461$	−4.74986e6	−1.04095	−0.520474	−	0.853878i	$-0.674245\pi$
$461$	−4.74986e6	−1.04095	−0.520474	+	0.853878i	$0.674245\pi$
$462$	− 2.02214e6i	− 0.440765i
$463$	1.38789e6i	0.300887i	0.988619	+	0.150443i	$0.0480701\pi$
$463$	1.38789e6i	0.300887i	−0.988619	+	0.150443i	$0.951930\pi$
$464$	9.51023e6	2.05067
$465$	0	0
$466$	2.31525e6	0.493893
$467$	− 880354.i	− 0.186795i	−0.995629	−	0.0933975i	$-0.970227\pi$
$467$	− 880354.i	− 0.186795i	0.995629	−	0.0933975i	$-0.0297727\pi$
$468$	− 731594.i	− 0.154403i
$469$	−66070.1	−0.0138699
$470$	0	0
$471$	−692835.	−0.143906
$472$	− 4.25536e6i	− 0.879187i
$473$	6.44868e6i	1.32531i
$474$	−1.28479e6	−0.262656
$475$	0	0
$476$	270431.	0.0547064
$477$	2.83043e6i	0.569582i
$478$	− 618309.i	− 0.123776i
$479$	9.35917e6	1.86380	0.931898	−	0.362720i	$-0.118152\pi$
$479$	9.35917e6	1.86380	0.931898	+	0.362720i	$0.118152\pi$
$480$	0	0
$481$	195621.	0.0385525
$482$	− 4.25681e6i	− 0.834578i
$483$	1.73216e6i	0.337846i
$484$	−4.74303e6	−0.920327
$485$	0	0
$486$	420440.	0.0807445
$487$	5.25097e6i	1.00327i	0.865080	+	0.501634i	$0.167268\pi$
$487$	5.25097e6i	1.00327i	−0.865080	+	0.501634i	$0.832732\pi$
$488$	3.68547e6i	0.700557i
$489$	2.87393e6	0.543506
$490$	0	0
$491$	−6.61595e6	−1.23848	−0.619239	−	0.785202i	$-0.712559\pi$
$491$	−6.61595e6	−1.23848	−0.619239	+	0.785202i	$0.712559\pi$
$492$	− 1.85484e6i	− 0.345456i
$493$	2.20568e6i	0.408720i
$494$	4.98634e6	0.919314
$495$	0	0
$496$	1.09483e6	0.199822
$497$	− 1.54513e6i	− 0.280591i
$498$	− 5.35721e6i	− 0.967978i
$499$	13840.9	0.00248835	0.00124418	−	0.999999i	$-0.499604\pi$
$499$	13840.9	0.00248835	0.00124418	+	0.999999i	$0.499604\pi$
$500$	0	0
$501$	5.20698e6	0.926811
$502$	− 4.18080e6i	− 0.740457i
$503$	796031.i	0.140285i	0.997537	+	0.0701423i	$0.0223453\pi$
$503$	796031.i	0.140285i	−0.997537	+	0.0701423i	$0.977655\pi$
$504$	−375942.	−0.0659242
$505$	0	0
$506$	1.80103e7	3.12713
$507$	− 1.24141e6i	− 0.214483i
$508$	2.97743e6i	0.511897i
$509$	8.91932e6	1.52594	0.762969	−	0.646435i	$-0.223741\pi$
$509$	8.91932e6	1.52594	0.762969	+	0.646435i	$0.223741\pi$
$510$	0	0
$511$	−517318.	−0.0876406
$512$	3.81817e6i	0.643695i
$513$	1.05683e6i	0.177301i
$514$	−7.57273e6	−1.26428
$515$	0	0
$516$	−1.68502e6	−0.278599
$517$	9.96333e6i	1.63937i
$518$	141283.i	0.0231348i
$519$	3.00487e6	0.489674
$520$	0	0
$521$	−4.98454e6	−0.804509	−0.402255	−	0.915528i	$-0.631773\pi$
$521$	−4.98454e6	−0.804509	−0.402255	+	0.915528i	$0.631773\pi$
$522$	4.30955e6i	0.692239i
$523$	107707.i	0.0172183i	0.999963	+	0.00860914i	$0.00274041\pi$
$523$	107707.i	0.0172183i	−0.999963	+	0.00860914i	$0.997260\pi$
$524$	5.55975e6	0.884560
$525$	0	0
$526$	−429769.	−0.0677284
$527$	253922.i	0.0398267i
$528$	7.37666e6i	1.15153i
$529$	−8.99119e6	−1.39694
$530$	0	0
$531$	3.63900e6	0.560074
$532$	1.32815e6i	0.203455i
$533$	5.32479e6i	0.811867i
$534$	6.53255e6	0.991356
$535$	0	0
$536$	127717.	0.0192016
$537$	1.08783e6i	0.162789i
$538$	− 8.38307e6i	− 1.24867i
$539$	−1.54623e6	−0.229246
$540$	0	0
$541$	6.59040e6	0.968096	0.484048	−	0.875041i	$-0.339166\pi$
$541$	6.59040e6	0.968096	0.484048	+	0.875041i	$0.339166\pi$
$542$	− 4.29423e6i	− 0.627896i
$543$	3.14798e6i	0.458175i
$544$	−1.78024e6	−0.257917
$545$	0	0
$546$	−1.51685e6	−0.217751
$547$	7.47564e6i	1.06827i	0.845400	+	0.534134i	$0.179362\pi$
$547$	7.47564e6i	1.06827i	−0.845400	+	0.534134i	$0.820638\pi$
$548$	2.38109e6i	0.338708i
$549$	−3.15165e6	−0.446280
$550$	0	0
$551$	−1.08326e7	−1.52004
$552$	− 3.34835e6i	− 0.467717i
$553$	982416.i	0.136610i
$554$	1.47517e7	2.04206
$555$	0	0
$556$	−6.73878e6	−0.924473
$557$	3.01966e6i	0.412401i	0.978510	+	0.206200i	$0.0661099\pi$
$557$	3.01966e6i	0.412401i	−0.978510	+	0.206200i	$0.933890\pi$
$558$	496123.i	0.0674535i
$559$	4.83728e6	0.654744
$560$	0	0
$561$	−1.71085e6	−0.229512
$562$	− 1.19647e7i	− 1.59794i
$563$	1.20403e7i	1.60090i	0.599397	+	0.800452i	$0.295407\pi$
$563$	1.20403e7i	1.60090i	−0.599397	+	0.800452i	$0.704593\pi$
$564$	−2.60338e6	−0.344619
$565$	0	0
$566$	−8.71919e6	−1.14402
$567$	− 321489.i	− 0.0419961i
$568$	2.98681e6i	0.388452i
$569$	6.55161e6	0.848335	0.424167	−	0.905584i	$-0.360567\pi$
$569$	6.55161e6	0.848335	0.424167	+	0.905584i	$0.360567\pi$
$570$	0	0
$571$	−5.08660e6	−0.652886	−0.326443	−	0.945217i	$-0.605850\pi$
$571$	−5.08660e6	−0.652886	−0.326443	+	0.945217i	$0.605850\pi$
$572$	5.81657e6i	0.743322i
$573$	− 3.84974e6i	− 0.489829i
$574$	−3.84572e6	−0.487190
$575$	0	0
$576$	−179393.	−0.0225294
$577$	1.50002e7i	1.87568i	0.347074	+	0.937838i	$0.387175\pi$
$577$	1.50002e7i	1.87568i	−0.347074	+	0.937838i	$0.612825\pi$
$578$	9.48925e6i	1.18144i
$579$	−8.95599e6	−1.11024
$580$	0	0
$581$	−4.09639e6	−0.503455
$582$	− 9.32415e6i	− 1.14104i
$583$	− 2.25035e7i	− 2.74207i
$584$	1.00000e6	0.121330
$585$	0	0
$586$	1.23201e7	1.48207
$587$	1.09548e7i	1.31223i	0.754663	+	0.656113i	$0.227800\pi$
$587$	1.09548e7i	1.31223i	−0.754663	+	0.656113i	$0.772200\pi$
$588$	− 404024.i	− 0.0481908i
$589$	−1.24707e6	−0.148116
$590$	0	0
$591$	2.80267e6	0.330067
$592$	− 515392.i	− 0.0604412i
$593$	1.57310e7i	1.83704i	0.395376	+	0.918519i	$0.370614\pi$
$593$	1.57310e7i	1.83704i	−0.395376	+	0.918519i	$0.629386\pi$
$594$	−3.34273e6	−0.388718
$595$	0	0
$596$	−1.93531e6	−0.223169
$597$	− 1.89712e6i	− 0.217851i
$598$	− 1.35099e7i	− 1.54489i
$599$	−1.28603e7	−1.46448	−0.732242	−	0.681044i	$-0.761526\pi$
$599$	−1.28603e7	−1.46448	−0.732242	+	0.681044i	$0.761526\pi$
$600$	0	0
$601$	−2.21487e6	−0.250128	−0.125064	−	0.992149i	$-0.539914\pi$
$601$	−2.21487e6	−0.250128	−0.125064	+	0.992149i	$0.539914\pi$
$602$	3.49362e6i	0.392903i
$603$	109218.i	0.0122321i
$604$	5.86044e6	0.653639
$605$	0	0
$606$	6.89899e6	0.763140
$607$	7.90528e6i	0.870854i	0.900224	+	0.435427i	$0.143403\pi$
$607$	7.90528e6i	0.870854i	−0.900224	+	0.435427i	$0.856597\pi$
$608$	− 8.74316e6i	− 0.959201i
$609$	3.29530e6	0.360041
$610$	0	0
$611$	7.47368e6	0.809900
$612$	− 447038.i	− 0.0482466i
$613$	4.40179e6i	0.473127i	0.971616	+	0.236564i	$0.0760212\pi$
$613$	4.40179e6i	0.473127i	−0.971616	+	0.236564i	$0.923979\pi$
$614$	4.61663e6	0.494202
$615$	0	0
$616$	2.98895e6	0.317370
$617$	1.23337e7i	1.30430i	0.758088	+	0.652152i	$0.226134\pi$
$617$	1.23337e7i	1.30430i	−0.758088	+	0.652152i	$0.773866\pi$
$618$	− 6.26478e6i	− 0.659834i
$619$	1.85120e7	1.94190	0.970949	−	0.239288i	$-0.0769141\pi$
$619$	1.85120e7	1.94190	0.970949	+	0.239288i	$0.0769141\pi$
$620$	0	0
$621$	2.86336e6	0.297952
$622$	− 4.57935e6i	− 0.474600i
$623$	− 4.99511e6i	− 0.515614i
$624$	5.53337e6	0.568890
$625$	0	0
$626$	−9.54278e6	−0.973283
$627$	− 8.40239e6i	− 0.853559i
$628$	1.43933e6i	0.145633i
$629$	119533.	0.0120466
$630$	0	0
$631$	−6.52547e6	−0.652437	−0.326219	−	0.945294i	$-0.605775\pi$
$631$	−6.52547e6	−0.652437	−0.326219	+	0.945294i	$0.605775\pi$
$632$	− 1.89906e6i	− 0.189124i
$633$	− 7.35753e6i	− 0.729831i
$634$	1.44013e6	0.142291
$635$	0	0
$636$	5.88007e6	0.576421
$637$	1.15986e6i	0.113255i
$638$	− 3.42633e7i	− 3.33256i
$639$	−2.55419e6	−0.247458
$640$	0	0
$641$	5.41110e6	0.520164	0.260082	−	0.965587i	$-0.416250\pi$
$641$	5.41110e6	0.520164	0.260082	+	0.965587i	$0.416250\pi$
$642$	− 1.27212e7i	− 1.21812i
$643$	− 1.88207e7i	− 1.79518i	−0.440831	−	0.897590i	$-0.645316\pi$
$643$	− 1.88207e7i	− 1.79518i	0.440831	−	0.897590i	$-0.354684\pi$
$644$	3.59846e6	0.341903
$645$	0	0
$646$	3.04689e6	0.287260
$647$	− 8.44010e6i	− 0.792660i	−0.918108	−	0.396330i	$-0.870284\pi$
$647$	− 8.44010e6i	− 0.792660i	0.918108	−	0.396330i	$-0.129716\pi$
$648$	621455.i	0.0581396i
$649$	−2.89320e7	−2.69629
$650$	0	0
$651$	379360.	0.0350832
$652$	− 5.97045e6i	− 0.550032i
$653$	− 1.72233e7i	− 1.58064i	−0.612692	−	0.790322i	$-0.709913\pi$
$653$	− 1.72233e7i	− 1.58064i	0.612692	−	0.790322i	$-0.290087\pi$
$654$	1.66997e6	0.152673
$655$	0	0
$656$	1.40290e7	1.27282
$657$	855158.i	0.0772917i
$658$	5.39771e6i	0.486010i
$659$	2.12784e7	1.90865	0.954324	−	0.298772i	$-0.0965771\pi$
$659$	2.12784e7	1.90865	0.954324	+	0.298772i	$0.0965771\pi$
$660$	0	0
$661$	−8.31227e6	−0.739973	−0.369987	−	0.929037i	$-0.620638\pi$
$661$	−8.31227e6	−0.739973	−0.369987	+	0.929037i	$0.620638\pi$
$662$	− 1.21938e6i	− 0.108142i
$663$	1.28334e6i	0.113386i
$664$	7.91854e6	0.696987
$665$	0	0
$666$	233549.	0.0204030
$667$	2.93497e7i	2.55440i
$668$	− 1.08172e7i	− 0.937939i
$669$	4.50807e6	0.389427
$670$	0	0
$671$	2.50574e7	2.14847
$672$	2.65968e6i	0.227199i
$673$	− 1.14784e7i	− 0.976883i	−0.872597	−	0.488442i	$-0.837566\pi$
$673$	− 1.14784e7i	− 0.976883i	0.872597	−	0.488442i	$-0.162434\pi$
$674$	8.07774e6	0.684920
$675$	0	0
$676$	−2.57896e6	−0.217059
$677$	− 6.18494e6i	− 0.518638i	−0.965792	−	0.259319i	$-0.916502\pi$
$677$	− 6.18494e6i	− 0.518638i	0.965792	−	0.259319i	$-0.0834980\pi$
$678$	− 8.55277e6i	− 0.714550i
$679$	−7.12971e6	−0.593468
$680$	0	0
$681$	2.77398e6	0.229211
$682$	− 3.94445e6i	− 0.324733i
$683$	− 1.40269e7i	− 1.15056i	−0.817955	−	0.575282i	$-0.804892\pi$
$683$	− 1.40269e7i	− 1.15056i	0.817955	−	0.575282i	$-0.195108\pi$
$684$	2.19551e6	0.179430
$685$	0	0
$686$	−837683.	−0.0679625
$687$	− 4.55575e6i	− 0.368272i
$688$	− 1.27445e7i	− 1.02648i
$689$	−1.68803e7	−1.35466
$690$	0	0
$691$	6.61940e6	0.527380	0.263690	−	0.964608i	$-0.415060\pi$
$691$	6.61940e6	0.527380	0.263690	+	0.964608i	$0.415060\pi$
$692$	− 6.24245e6i	− 0.495553i
$693$	2.55601e6i	0.202176i
$694$	−3.81730e6	−0.300855
$695$	0	0
$696$	−6.36998e6	−0.498443
$697$	3.25370e6i	0.253685i
$698$	1.91628e7i	1.48875i
$699$	−2.92650e6	−0.226546
$700$	0	0
$701$	−1.30680e7	−1.00442	−0.502208	−	0.864747i	$-0.667479\pi$
$701$	−1.30680e7	−1.00442	−0.502208	+	0.864747i	$0.667479\pi$
$702$	2.50744e6i	0.192039i
$703$	587057.i	0.0448014i
$704$	1.42627e6	0.108460
$705$	0	0
$706$	−1.54483e7	−1.16645
$707$	− 5.27531e6i	− 0.396917i
$708$	− 7.55983e6i	− 0.566799i
$709$	−929866.	−0.0694712	−0.0347356	−	0.999397i	$-0.511059\pi$
$709$	−929866.	−0.0694712	−0.0347356	+	0.999397i	$0.511059\pi$
$710$	0	0
$711$	1.62399e6	0.120479
$712$	9.65581e6i	0.713820i
$713$	3.37879e6i	0.248907i
$714$	−926866.	−0.0680411
$715$	0	0
$716$	2.25991e6	0.164743
$717$	781550.i	0.0567752i
$718$	2.43640e7i	1.76376i
$719$	−8.51260e6	−0.614101	−0.307051	−	0.951693i	$-0.599342\pi$
$719$	−8.51260e6	−0.614101	−0.307051	+	0.951693i	$0.599342\pi$
$720$	0	0
$721$	−4.79036e6	−0.343186
$722$	− 2.66631e6i	− 0.190356i
$723$	5.38066e6i	0.382816i
$724$	6.53976e6	0.463677
$725$	0	0
$726$	1.62561e7	1.14466
$727$	− 1.81831e7i	− 1.27595i	−0.770059	−	0.637973i	$-0.779773\pi$
$727$	− 1.81831e7i	− 1.27595i	0.770059	−	0.637973i	$-0.220227\pi$
$728$	− 2.24206e6i	− 0.156790i
$729$	−531441.	−0.0370370
$730$	0	0
$731$	2.95581e6	0.204589
$732$	6.54740e6i	0.451638i
$733$	− 9.79622e6i	− 0.673440i	−0.941605	−	0.336720i	$-0.890683\pi$
$733$	− 9.79622e6i	− 0.673440i	0.941605	−	0.336720i	$-0.109317\pi$
$734$	1.18673e7	0.813040
$735$	0	0
$736$	−2.36886e7	−1.61192
$737$	− 868343.i	− 0.0588874i
$738$	6.35721e6i	0.429661i
$739$	5.55814e6	0.374385	0.187192	−	0.982323i	$-0.440061\pi$
$739$	5.55814e6	0.374385	0.187192	+	0.982323i	$0.440061\pi$
$740$	0	0
$741$	−6.30279e6	−0.421684
$742$	− 1.21914e7i	− 0.812914i
$743$	3.01729e6i	0.200514i	0.994962	+	0.100257i	$0.0319666\pi$
$743$	3.01729e6i	0.200514i	−0.994962	+	0.100257i	$0.968033\pi$
$744$	−733324.	−0.0485695
$745$	0	0
$746$	2.43269e7	1.60044
$747$	6.77158e6i	0.444006i
$748$	3.55420e6i	0.232267i
$749$	−9.72722e6	−0.633555
$750$	0	0
$751$	2.95462e7	1.91162	0.955810	−	0.293985i	$-0.0949815\pi$
$751$	2.95462e7	1.91162	0.955810	+	0.293985i	$0.0949815\pi$
$752$	− 1.96905e7i	− 1.26973i
$753$	5.28458e6i	0.339643i
$754$	−2.57016e7	−1.64638
$755$	0	0
$756$	−667877.	−0.0425003
$757$	− 2.43622e7i	− 1.54517i	−0.634911	−	0.772585i	$-0.718963\pi$
$757$	− 2.43622e7i	− 1.54517i	0.634911	−	0.772585i	$-0.281037\pi$
$758$	3.01983e6i	0.190902i
$759$	−2.27653e7	−1.43439
$760$	0	0
$761$	7.25049e6	0.453843	0.226921	−	0.973913i	$-0.427134\pi$
$761$	7.25049e6	0.453843	0.226921	+	0.973913i	$0.427134\pi$
$762$	− 1.02048e7i	− 0.636672i
$763$	− 1.27694e6i	− 0.0794069i
$764$	−7.99763e6	−0.495710
$765$	0	0
$766$	1.75187e7	1.07877
$767$	2.17025e7i	1.33205i
$768$	− 1.19946e7i	− 0.733809i
$769$	−3.98577e6	−0.243051	−0.121525	−	0.992588i	$-0.538779\pi$
$769$	−3.98577e6	−0.243051	−0.121525	+	0.992588i	$0.538779\pi$
$770$	0	0
$771$	9.57203e6	0.579920
$772$	1.86056e7i	1.12357i
$773$	1.47149e7i	0.885745i	0.896585	+	0.442872i	$0.146040\pi$
$773$	1.47149e7i	0.885745i	−0.896585	+	0.442872i	$0.853960\pi$
$774$	5.77517e6	0.346507
$775$	0	0
$776$	1.37821e7	0.821601
$777$	− 178583.i	− 0.0106118i
$778$	− 1.45928e7i	− 0.864347i
$779$	−1.59797e7	−0.943462
$780$	0	0
$781$	2.03072e7	1.19130
$782$	− 8.25518e6i	− 0.482736i
$783$	− 5.44733e6i	− 0.317526i
$784$	3.05582e6	0.177557
$785$	0	0
$786$	−1.90553e7	−1.10017
$787$	6.49761e6i	0.373953i	0.982364	+	0.186976i	$0.0598688\pi$
$787$	6.49761e6i	0.373953i	−0.982364	+	0.186976i	$0.940131\pi$
$788$	− 5.82239e6i	− 0.334030i
$789$	543233.	0.0310666
$790$	0	0
$791$	−6.53987e6	−0.371645
$792$	− 4.94091e6i	− 0.279894i
$793$	− 1.87960e7i	− 1.06141i
$794$	1.34156e7	0.755197
$795$	0	0
$796$	−3.94117e6	−0.220466
$797$	1.68451e6i	0.0939352i	0.998896	+	0.0469676i	$0.0149558\pi$
$797$	1.68451e6i	0.0939352i	−0.998896	+	0.0469676i	$0.985044\pi$
$798$	− 4.55206e6i	− 0.253047i
$799$	4.56677e6	0.253071
$800$	0	0
$801$	−8.25722e6	−0.454729
$802$	− 1.93814e7i	− 1.06402i
$803$	− 6.79898e6i	− 0.372096i
$804$	226895.	0.0123790
$805$	0	0
$806$	−2.95881e6	−0.160428
$807$	1.05963e7i	0.572757i
$808$	1.01975e7i	0.549494i
$809$	1.39278e7	0.748189	0.374094	−	0.927391i	$-0.377954\pi$
$809$	1.39278e7	0.748189	0.374094	+	0.927391i	$0.377954\pi$
$810$	0	0
$811$	−1.60742e7	−0.858180	−0.429090	−	0.903262i	$-0.641166\pi$
$811$	−1.60742e7	−0.858180	−0.429090	+	0.903262i	$0.641166\pi$
$812$	− 6.84581e6i	− 0.364363i
$813$	5.42796e6i	0.288012i
$814$	−1.85685e6	−0.0982234
$815$	0	0
$816$	3.38115e6	0.177762
$817$	1.45167e7i	0.760872i
$818$	− 4.77958e7i	− 2.49750i
$819$	1.91731e6	0.0998812
$820$	0	0
$821$	2.78915e7	1.44416	0.722079	−	0.691811i	$-0.243187\pi$
$821$	2.78915e7	1.44416	0.722079	+	0.691811i	$0.243187\pi$
$822$	− 8.16089e6i	− 0.421267i
$823$	1.00664e7i	0.518053i	0.965870	+	0.259027i	$0.0834018\pi$
$823$	1.00664e7i	0.518053i	−0.965870	+	0.259027i	$0.916598\pi$
$824$	9.26002e6	0.475110
$825$	0	0
$826$	−1.56742e7	−0.799344
$827$	2.13232e7i	1.08415i	0.840331	+	0.542073i	$0.182360\pi$
$827$	2.13232e7i	1.08415i	−0.840331	+	0.542073i	$0.817640\pi$
$828$	− 5.94848e6i	− 0.301530i
$829$	−9.12162e6	−0.460984	−0.230492	−	0.973074i	$-0.574033\pi$
$829$	−9.12162e6	−0.460984	−0.230492	+	0.973074i	$0.574033\pi$
$830$	0	0
$831$	−1.86463e7	−0.936680
$832$	− 1.06988e6i	− 0.0535827i
$833$	708727.i	0.0353889i
$834$	2.30963e7	1.14981
$835$	0	0
$836$	−1.74555e7	−0.863808
$837$	− 627106.i	− 0.0309405i
$838$	− 2.00597e7i	− 0.986765i
$839$	−1.63850e7	−0.803602	−0.401801	−	0.915727i	$-0.631616\pi$
$839$	−1.63850e7	−0.803602	−0.401801	+	0.915727i	$0.631616\pi$
$840$	0	0
$841$	3.53246e7	1.72221
$842$	3.96086e7i	1.92535i
$843$	1.51235e7i	0.732964i
$844$	−1.52849e7	−0.738594
$845$	0	0
$846$	8.92274e6	0.428620
$847$	− 1.24302e7i	− 0.595347i
$848$	4.44736e7i	2.12379i
$849$	1.10212e7	0.524757
$850$	0	0
$851$	1.59056e6	0.0752882
$852$	5.30620e6i	0.250429i
$853$	1.49098e7i	0.701615i	0.936448	+	0.350808i	$0.114093\pi$
$853$	1.49098e7i	0.701615i	−0.936448	+	0.350808i	$0.885907\pi$
$854$	1.35750e7	0.636936
$855$	0	0
$856$	1.88032e7	0.877098
$857$	2.00613e6i	0.0933053i	0.998911	+	0.0466526i	$0.0148554\pi$
$857$	2.00613e6i	0.0933053i	−0.998911	+	0.0466526i	$0.985145\pi$
$858$	− 1.99356e7i	− 0.924507i
$859$	2.45641e7	1.13584	0.567921	−	0.823083i	$-0.307748\pi$
$859$	2.45641e7	1.13584	0.567921	+	0.823083i	$0.307748\pi$
$860$	0	0
$861$	4.86104e6	0.223471
$862$	9.73279e6i	0.446138i
$863$	1.98157e7i	0.905697i	0.891588	+	0.452848i	$0.149592\pi$
$863$	1.98157e7i	0.905697i	−0.891588	+	0.452848i	$0.850408\pi$
$864$	4.39661e6	0.200371
$865$	0	0
$866$	1.07527e7	0.487219
$867$	− 1.19945e7i	− 0.541920i
$868$	− 788102.i	− 0.0355045i
$869$	−1.29116e7	−0.580005
$870$	0	0
$871$	−651361.	−0.0290922
$872$	2.46839e6i	0.109932i
$873$	1.17858e7i	0.523390i
$874$	4.05431e7	1.79531
$875$	0	0
$876$	1.77655e6	0.0782198
$877$	− 2.03905e7i	− 0.895218i	−0.894229	−	0.447609i	$-0.852276\pi$
$877$	− 2.03905e7i	− 0.895218i	0.894229	−	0.447609i	$-0.147724\pi$
$878$	1.90568e7i	0.834282i
$879$	−1.55727e7	−0.679816
$880$	0	0
$881$	−3.33889e7	−1.44931	−0.724656	−	0.689111i	$-0.758001\pi$
$881$	−3.33889e7	−1.44931	−0.724656	+	0.689111i	$0.758001\pi$
$882$	1.38474e6i	0.0599373i
$883$	− 1.06634e7i	− 0.460251i	−0.973161	−	0.230126i	$-0.926086\pi$
$883$	− 1.06634e7i	− 0.460251i	0.973161	−	0.230126i	$-0.0739137\pi$
$884$	2.66607e6	0.114747
$885$	0	0
$886$	4.98953e7	2.13538
$887$	1.26547e7i	0.540062i	0.962852	+	0.270031i	$0.0870339\pi$
$887$	1.26547e7i	0.540062i	−0.962852	+	0.270031i	$0.912966\pi$
$888$	345211.i	0.0146910i
$889$	−7.80307e6	−0.331139
$890$	0	0
$891$	4.22525e6	0.178303
$892$	− 9.36529e6i	− 0.394102i
$893$	2.24285e7i	0.941177i
$894$	6.63302e6	0.277567
$895$	0	0
$896$	−8.68394e6	−0.361366
$897$	1.70767e7i	0.708634i
$898$	7.72875e6i	0.319829i
$899$	6.42790e6	0.265259
$900$	0	0
$901$	−1.03146e7	−0.423294
$902$	− 5.05433e7i	− 2.06846i
$903$	− 4.41598e6i	− 0.180222i
$904$	1.26419e7	0.514508
$905$	0	0
$906$	−2.00859e7	−0.812963
$907$	3.06856e7i	1.23856i	0.785171	+	0.619279i	$0.212575\pi$
$907$	3.06856e7i	1.23856i	−0.785171	+	0.619279i	$0.787425\pi$
$908$	− 5.76281e6i	− 0.231963i
$909$	−8.72041e6	−0.350048
$910$	0	0
$911$	−2.66726e7	−1.06480	−0.532402	−	0.846492i	$-0.678710\pi$
$911$	−2.66726e7	−1.06480	−0.532402	+	0.846492i	$0.678710\pi$
$912$	1.66056e7i	0.661102i
$913$	− 5.38378e7i	− 2.13752i
$914$	−5.30589e7	−2.10084
$915$	0	0
$916$	−9.46434e6	−0.372693
$917$	1.45706e7i	0.572210i
$918$	1.53217e6i	0.0600066i
$919$	−3.26203e7	−1.27409	−0.637045	−	0.770827i	$-0.719843\pi$
$919$	−3.26203e7	−1.27409	−0.637045	+	0.770827i	$0.719843\pi$
$920$	0	0
$921$	−5.83548e6	−0.226688
$922$	− 3.38199e7i	− 1.31022i
$923$	− 1.52328e7i	− 0.588540i
$924$	5.30999e6	0.204604
$925$	0	0
$926$	−9.88204e6	−0.378721
$927$	7.91876e6i	0.302662i
$928$	4.50658e7i	1.71782i
$929$	−3.87078e7	−1.47150	−0.735748	−	0.677256i	$-0.763169\pi$
$929$	−3.87078e7	−1.47150	−0.735748	+	0.677256i	$0.763169\pi$
$930$	0	0
$931$	−3.48073e6	−0.131612
$932$	6.07965e6i	0.229266i
$933$	5.78836e6i	0.217696i
$934$	6.26828e6	0.235116
$935$	0	0
$936$	−3.70627e6	−0.138276
$937$	1.38273e7i	0.514502i	0.966345	+	0.257251i	$0.0828167\pi$
$937$	1.38273e7i	0.514502i	−0.966345	+	0.257251i	$0.917183\pi$
$938$	− 470432.i	− 0.0174578i
$939$	1.20622e7	0.446439
$940$	0	0
$941$	3.07596e7	1.13242	0.566208	−	0.824262i	$-0.308410\pi$
$941$	3.07596e7	1.13242	0.566208	+	0.824262i	$0.308410\pi$
$942$	− 4.93311e6i	− 0.181131i
$943$	4.32951e7i	1.58547i
$944$	5.71784e7	2.08834
$945$	0	0
$946$	−4.59158e7	−1.66815
$947$	− 9.78578e6i	− 0.354585i	−0.984158	−	0.177293i	$-0.943266\pi$
$947$	− 9.78578e6i	− 0.354585i	0.984158	−	0.177293i	$-0.0567339\pi$
$948$	− 3.37376e6i	− 0.121925i
$949$	−5.10004e6	−0.183827
$950$	0	0
$951$	−1.82034e6	−0.0652680
$952$	− 1.37001e6i	− 0.0489926i
$953$	− 1.31109e7i	− 0.467627i	−0.972282	−	0.233813i	$-0.924880\pi$
$953$	− 1.31109e7i	− 0.467627i	0.972282	−	0.233813i	$-0.0751205\pi$
$954$	−2.01532e7	−0.716923
$955$	0	0
$956$	1.62363e6	0.0574569
$957$	4.33093e7i	1.52862i
$958$	6.66390e7i	2.34593i
$959$	−6.24022e6	−0.219105
$960$	0	0
$961$	−2.78892e7	−0.974153
$962$	1.39286e6i	0.0485253i
$963$	1.60797e7i	0.558743i
$964$	1.11780e7	0.387412
$965$	0	0
$966$	−1.23333e7	−0.425241
$967$	− 1.90037e7i	− 0.653540i	−0.945104	−	0.326770i	$-0.894040\pi$
$967$	− 1.90037e7i	− 0.653540i	0.945104	−	0.326770i	$-0.105960\pi$
$968$	2.40283e7i	0.824203i
$969$	−3.85130e6	−0.131764
$970$	0	0
$971$	−2.27846e7	−0.775522	−0.387761	−	0.921760i	$-0.626751\pi$
$971$	−2.27846e7	−0.775522	−0.387761	+	0.921760i	$0.626751\pi$
$972$	1.10404e6i	0.0374817i
$973$	− 1.76606e7i	− 0.598029i
$974$	−3.73879e7	−1.26280
$975$	0	0
$976$	−4.95209e7	−1.66404
$977$	2.72101e7i	0.911999i	0.889980	+	0.456000i	$0.150718\pi$
$977$	2.72101e7i	0.911999i	−0.889980	+	0.456000i	$0.849282\pi$
$978$	2.04629e7i	0.684102i
$979$	6.56494e7	2.18914
$980$	0	0
$981$	−2.11086e6	−0.0700303
$982$	− 4.71068e7i	− 1.55885i
$983$	− 4.20145e7i	− 1.38681i	−0.720550	−	0.693403i	$-0.756111\pi$
$983$	− 4.20145e7i	− 1.38681i	0.720550	−	0.693403i	$-0.243889\pi$
$984$	−9.39664e6	−0.309375
$985$	0	0
$986$	−1.57049e7	−0.514449
$987$	− 6.82277e6i	− 0.222930i
$988$	1.30937e7i	0.426747i
$989$	3.93312e7	1.27863
$990$	0	0
$991$	−9.17151e6	−0.296658	−0.148329	−	0.988938i	$-0.547390\pi$
$991$	−9.17151e6	−0.296658	−0.148329	+	0.988938i	$0.547390\pi$
$992$	5.18805e6i	0.167388i
$993$	1.54131e6i	0.0496040i
$994$	1.10016e7	0.353175
$995$	0	0
$996$	1.40676e7	0.449337
$997$	2.48701e7i	0.792392i	0.918166	+	0.396196i	$0.129670\pi$
$997$	2.48701e7i	0.792392i	−0.918166	+	0.396196i	$0.870330\pi$
$998$	98549.5i	0.00313205i
$999$	−295209.	−0.00935872

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.6.d.p.274.9		12
5.2	odd	4		525.6.a.r.1.2	✓	6
5.3	odd	4		525.6.a.s.1.5	yes	6
5.4	even	2	inner	525.6.d.p.274.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.6.a.r.1.2	✓	6	5.2	odd	4
525.6.a.s.1.5	yes	6	5.3	odd	4
525.6.d.p.274.4		12	5.4	even	2	inner
525.6.d.p.274.9		12	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:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	525.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+7.12018i q^{2} -9.00000i q^{3} -18.6970 q^{4} +64.0817 q^{6} -49.0000i q^{7} +94.7196i q^{8} -81.0000 q^{9} +O(q^{10})\)	\(q+7.12018i q^{2} -9.00000i q^{3} -18.6970 q^{4} +64.0817 q^{6} -49.0000i q^{7} +94.7196i q^{8} -81.0000 q^{9} +643.995 q^{11} +168.273i q^{12} -483.073i q^{13} +348.889 q^{14} -1272.73 q^{16} -295.180i q^{17} -576.735i q^{18} +1449.70 q^{19} -441.000 q^{21} +4585.36i q^{22} -3927.79i q^{23} +852.476 q^{24} +3439.57 q^{26} +729.000i q^{27} +916.155i q^{28} -7472.33 q^{29} -860.228 q^{31} -6031.02i q^{32} -5795.95i q^{33} +2101.74 q^{34} +1514.46 q^{36} +404.951i q^{37} +10322.1i q^{38} -4347.65 q^{39} -11022.8 q^{41} -3140.00i q^{42} +10013.6i q^{43} -12040.8 q^{44} +27966.6 q^{46} +15471.1i q^{47} +11454.5i q^{48} -2401.00 q^{49} -2656.62 q^{51} +9032.02i q^{52} -34943.6i q^{53} -5190.61 q^{54} +4641.26 q^{56} -13047.3i q^{57} -53204.4i q^{58} -44925.9 q^{59} +38909.3 q^{61} -6124.98i q^{62} +3969.00i q^{63} +2214.73 q^{64} +41268.2 q^{66} -1348.37i q^{67} +5518.99i q^{68} -35350.1 q^{69} +31533.2 q^{71} -7672.29i q^{72} -10557.5i q^{73} -2883.33 q^{74} -27105.1 q^{76} -31555.7i q^{77} -30956.1i q^{78} -20049.3 q^{79} +6561.00 q^{81} -78484.1i q^{82} -83599.8i q^{83} +8245.39 q^{84} -71298.4 q^{86} +67251.0i q^{87} +60998.9i q^{88} +101941. q^{89} -23670.6 q^{91} +73438.0i q^{92} +7742.05i q^{93} -110157. q^{94} -54279.2 q^{96} -145504. i q^{97} -17095.6i q^{98} -52163.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 264 q^{4} + 72 q^{6} - 972 q^{9}+O(q^{10}) \) 12 * q - 264 * q^4 + 72 * q^6 - 972 * q^9	\( 12 q - 264 q^{4} + 72 q^{6} - 972 q^{9} + 392 q^{14} + 3568 q^{16} - 520 q^{19} - 5292 q^{21} - 5454 q^{24} - 6534 q^{26} + 4244 q^{29} + 4524 q^{31} - 24778 q^{34} + 21384 q^{36} + 3816 q^{39} + 16248 q^{41} - 62322 q^{44} + 58720 q^{46} - 28812 q^{49} + 67320 q^{51} - 5832 q^{54} - 29694 q^{56} - 51316 q^{59} + 82376 q^{61} - 131894 q^{64} + 124218 q^{66} + 33336 q^{69} + 35160 q^{71} - 7942 q^{74} + 507092 q^{76} + 19472 q^{79} + 78732 q^{81} + 116424 q^{84} + 287632 q^{86} + 660720 q^{89} + 20776 q^{91} + 747360 q^{94} + 491220 q^{96}+O(q^{100}) \) 12 * q - 264 * q^4 + 72 * q^6 - 972 * q^9 + 392 * q^14 + 3568 * q^16 - 520 * q^19 - 5292 * q^21 - 5454 * q^24 - 6534 * q^26 + 4244 * q^29 + 4524 * q^31 - 24778 * q^34 + 21384 * q^36 + 3816 * q^39 + 16248 * q^41 - 62322 * q^44 + 58720 * q^46 - 28812 * q^49 + 67320 * q^51 - 5832 * q^54 - 29694 * q^56 - 51316 * q^59 + 82376 * q^61 - 131894 * q^64 + 124218 * q^66 + 33336 * q^69 + 35160 * q^71 - 7942 * q^74 + 507092 * q^76 + 19472 * q^79 + 78732 * q^81 + 116424 * q^84 + 287632 * q^86 + 660720 * q^89 + 20776 * q^91 + 747360 * q^94 + 491220 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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