LMFDB - Embedded newform 325.6.b.h.274.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(274,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("325.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.b (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:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 378 x^{16} + 59175 x^{14} + 4956036 x^{12} + 239061247 x^{10} + 6629044458 x^{8} + \cdots + 134659641600 $ x^18 + 378x^16 + 59175x^14 + 4956036x^12 + 239061247x^10 + 6629044458x^8 + 98240243849x^6 + 625586966104x^4 + 584919029904x^2 + 134659641600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.9
Root			$0.603392i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.6.b.h.274.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.396608i q^{2} -11.4974i q^{3} +31.8427 q^{4} -4.55994 q^{6} -8.04695i q^{7} -25.3205i q^{8} +110.811 q^{9} +O(q^{10})$	\(q-0.396608i q^{2} -11.4974i q^{3} +31.8427 q^{4} -4.55994 q^{6} -8.04695i q^{7} -25.3205i q^{8} +110.811 q^{9} +335.903 q^{11} -366.107i q^{12} -169.000i q^{13} -3.19148 q^{14} +1008.92 q^{16} -38.1592i q^{17} -43.9484i q^{18} -1178.86 q^{19} -92.5186 q^{21} -133.222i q^{22} +1679.35i q^{23} -291.119 q^{24} -67.0267 q^{26} -4067.89i q^{27} -256.237i q^{28} +6705.42 q^{29} +7858.95 q^{31} -1210.40i q^{32} -3861.99i q^{33} -15.1342 q^{34} +3528.51 q^{36} +1715.81i q^{37} +467.545i q^{38} -1943.05 q^{39} -10076.9 q^{41} +36.6936i q^{42} +10822.3i q^{43} +10696.0 q^{44} +666.042 q^{46} -12999.2i q^{47} -11600.0i q^{48} +16742.2 q^{49} -438.730 q^{51} -5381.42i q^{52} +2639.19i q^{53} -1613.36 q^{54} -203.753 q^{56} +13553.8i q^{57} -2659.42i q^{58} -4956.81 q^{59} -43787.4 q^{61} -3116.92i q^{62} -891.687i q^{63} +31805.5 q^{64} -1531.70 q^{66} +9170.90i q^{67} -1215.09i q^{68} +19308.0 q^{69} -13030.5 q^{71} -2805.78i q^{72} -41726.8i q^{73} +680.503 q^{74} -37538.1 q^{76} -2702.99i q^{77} +770.630i q^{78} +49809.7 q^{79} -19843.0 q^{81} +3996.58i q^{82} -86196.9i q^{83} -2946.04 q^{84} +4292.19 q^{86} -77094.6i q^{87} -8505.22i q^{88} +67287.2 q^{89} -1359.93 q^{91} +53474.9i q^{92} -90357.2i q^{93} -5155.58 q^{94} -13916.4 q^{96} -175537. i q^{97} -6640.10i q^{98} +37221.6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9}+O(q^{10}) $ 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9	$ 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9} - 2844 q^{11} + 684 q^{14} - 2122 q^{16} + 816 q^{19} - 7824 q^{21} + 16938 q^{24} - 1690 q^{26} + 17474 q^{29} + 1496 q^{31} + 35578 q^{34} + 1024 q^{36} + 3718 q^{39} - 57352 q^{41} + 61198 q^{44} - 24112 q^{46} + 81814 q^{49} - 62012 q^{51} + 205522 q^{54} - 46004 q^{56} + 176284 q^{59} + 56330 q^{61} + 201690 q^{64} + 85154 q^{66} + 363494 q^{69} - 141124 q^{71} + 271352 q^{74} + 92746 q^{76} + 328146 q^{79} - 139870 q^{81} + 691312 q^{84} - 589840 q^{86} + 505396 q^{89} + 4056 q^{91} + 1003212 q^{94} - 638362 q^{96} + 1515552 q^{99}+O(q^{100}) $ 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9 - 2844 * q^11 + 684 * q^14 - 2122 * q^16 + 816 * q^19 - 7824 * q^21 + 16938 * q^24 - 1690 * q^26 + 17474 * q^29 + 1496 * q^31 + 35578 * q^34 + 1024 * q^36 + 3718 * q^39 - 57352 * q^41 + 61198 * q^44 - 24112 * q^46 + 81814 * q^49 - 62012 * q^51 + 205522 * q^54 - 46004 * q^56 + 176284 * q^59 + 56330 * q^61 + 201690 * q^64 + 85154 * q^66 + 363494 * q^69 - 141124 * q^71 + 271352 * q^74 + 92746 * q^76 + 328146 * q^79 - 139870 * q^81 + 691312 * q^84 - 589840 * q^86 + 505396 * q^89 + 4056 * q^91 + 1003212 * q^94 - 638362 * q^96 + 1515552 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 0.396608i	− 0.0701110i	−0.999385	−	0.0350555i	$-0.988839\pi$
$2$	− 0.396608i	− 0.0701110i	0.999385	−	0.0350555i	$-0.0111608\pi$
$3$	− 11.4974i	− 0.737556i	−0.929517	−	0.368778i	$-0.879776\pi$
$3$	− 11.4974i	− 0.737556i	0.929517	−	0.368778i	$-0.120224\pi$
$4$	31.8427	0.995084
$5$	0	0
$5$	0	0
$6$	−4.55994	−0.0517108
$7$	− 8.04695i	− 0.0620706i	−0.999518	−	0.0310353i	$-0.990120\pi$
$7$	− 8.04695i	− 0.0620706i	0.999518	−	0.0310353i	$-0.00988043\pi$
$8$	− 25.3205i	− 0.139877i
$9$	110.811	0.456011
$10$	0	0
$11$	335.903	0.837012	0.418506	−	0.908214i	$-0.362554\pi$
$11$	335.903	0.837012	0.418506	+	0.908214i	$0.362554\pi$
$12$	− 366.107i	− 0.733931i
$13$	− 169.000i	− 0.277350i
$13$	− 169.000i	− 0.277350i
$14$	−3.19148	−0.00435183
$15$	0	0
$16$	1008.92	0.985278
$17$	− 38.1592i	− 0.0320241i	−0.999872	−	0.0160121i	$-0.994903\pi$
$17$	− 38.1592i	− 0.0320241i	0.999872	−	0.0160121i	$-0.00509702\pi$
$18$	− 43.9484i	− 0.0319714i
$19$	−1178.86	−0.749167	−0.374584	−	0.927193i	$-0.622214\pi$
$19$	−1178.86	−0.749167	−0.374584	+	0.927193i	$0.622214\pi$
$20$	0	0
$21$	−92.5186	−0.0457806
$22$	− 133.222i	− 0.0586837i
$23$	1679.35i	0.661943i	0.943641	+	0.330972i	$0.107376\pi$
$23$	1679.35i	0.661943i	−0.943641	+	0.330972i	$0.892624\pi$
$24$	−291.119	−0.103167
$25$	0	0
$26$	−67.0267	−0.0194453
$27$	− 4067.89i	− 1.07389i
$28$	− 256.237i	− 0.0617655i
$29$	6705.42	1.48058	0.740288	−	0.672290i	$-0.234689\pi$
$29$	6705.42	1.48058	0.740288	+	0.672290i	$0.234689\pi$
$30$	0	0
$31$	7858.95	1.46879	0.734396	−	0.678721i	$-0.237465\pi$
$31$	7858.95	1.46879	0.734396	+	0.678721i	$0.237465\pi$
$32$	− 1210.40i	− 0.208956i
$33$	− 3861.99i	− 0.617343i
$34$	−15.1342	−0.00224524
$35$	0	0
$36$	3528.51	0.453769
$37$	1715.81i	0.206046i	0.994679	+	0.103023i	$0.0328516\pi$
$37$	1715.81i	0.206046i	−0.994679	+	0.103023i	$0.967148\pi$
$38$	467.545i	0.0525249i
$39$	−1943.05	−0.204561
$40$	0	0
$41$	−10076.9	−0.936198	−0.468099	−	0.883676i	$-0.655061\pi$
$41$	−10076.9	−0.936198	−0.468099	+	0.883676i	$0.655061\pi$
$42$	36.6936i	0.00320972i
$43$	10822.3i	0.892579i	0.894889	+	0.446289i	$0.147255\pi$
$43$	10822.3i	0.892579i	−0.894889	+	0.446289i	$0.852745\pi$
$44$	10696.0	0.832897
$45$	0	0
$46$	666.042	0.0464095
$47$	− 12999.2i	− 0.858364i	−0.903218	−	0.429182i	$-0.858802\pi$
$47$	− 12999.2i	− 0.858364i	0.903218	−	0.429182i	$-0.141198\pi$
$48$	− 11600.0i	− 0.726697i
$49$	16742.2	0.996147
$50$	0	0
$51$	−438.730	−0.0236196
$52$	− 5381.42i	− 0.275987i
$53$	2639.19i	0.129057i	0.997916	+	0.0645285i	$0.0205543\pi$
$53$	2639.19i	0.129057i	−0.997916	+	0.0645285i	$0.979446\pi$
$54$	−1613.36	−0.0752915
$55$	0	0
$56$	−203.753	−0.00868227
$57$	13553.8i	0.552553i
$58$	− 2659.42i	− 0.103805i
$59$	−4956.81	−0.185384	−0.0926919	−	0.995695i	$-0.529547\pi$
$59$	−4956.81	−0.185384	−0.0926919	+	0.995695i	$0.529547\pi$
$60$	0	0
$61$	−43787.4	−1.50669	−0.753346	−	0.657625i	$-0.771561\pi$
$61$	−43787.4	−1.50669	−0.753346	+	0.657625i	$0.771561\pi$
$62$	− 3116.92i	− 0.102979i
$63$	− 891.687i	− 0.0283049i
$64$	31805.5	0.970627
$65$	0	0
$66$	−1531.70	−0.0432825
$67$	9170.90i	0.249589i	0.992183	+	0.124794i	$0.0398271\pi$
$67$	9170.90i	0.249589i	−0.992183	+	0.124794i	$0.960173\pi$
$68$	− 1215.09i	− 0.0318667i
$69$	19308.0	0.488220
$70$	0	0
$71$	−13030.5	−0.306772	−0.153386	−	0.988166i	$-0.549018\pi$
$71$	−13030.5	−0.306772	−0.153386	+	0.988166i	$0.549018\pi$
$72$	− 2805.78i	− 0.0637856i
$73$	− 41726.8i	− 0.916447i	−0.888837	−	0.458224i	$-0.848486\pi$
$73$	− 41726.8i	− 0.916447i	0.888837	−	0.458224i	$-0.151514\pi$
$74$	680.503	0.0144461
$75$	0	0
$76$	−37538.1	−0.745485
$77$	− 2702.99i	− 0.0519538i
$78$	770.630i	0.0143420i
$79$	49809.7	0.897938	0.448969	−	0.893547i	$-0.351791\pi$
$79$	49809.7	0.897938	0.448969	+	0.893547i	$0.351791\pi$
$80$	0	0
$81$	−19843.0	−0.336043
$82$	3996.58i	0.0656378i
$83$	− 86196.9i	− 1.37340i	−0.726942	−	0.686699i	$-0.759059\pi$
$83$	− 86196.9i	− 1.37340i	0.726942	−	0.686699i	$-0.240941\pi$
$84$	−2946.04	−0.0455555
$85$	0	0
$86$	4292.19	0.0625796
$87$	− 77094.6i	− 1.09201i
$88$	− 8505.22i	− 0.117079i
$89$	67287.2	0.900446	0.450223	−	0.892916i	$-0.351344\pi$
$89$	67287.2	0.900446	0.450223	+	0.892916i	$0.351344\pi$
$90$	0	0
$91$	−1359.93	−0.0172153
$92$	53474.9i	0.658689i
$93$	− 90357.2i	− 1.08332i
$94$	−5155.58	−0.0601808
$95$	0	0
$96$	−13916.4	−0.154117
$97$	− 175537.i	− 1.89426i	−0.320850	−	0.947130i	$-0.603969\pi$
$97$	− 175537.i	− 1.89426i	0.320850	−	0.947130i	$-0.396031\pi$
$98$	− 6640.10i	− 0.0698409i
$99$	37221.6	0.381687
$100$	0	0
$101$	−20550.4	−0.200455	−0.100227	−	0.994965i	$-0.531957\pi$
$101$	−20550.4	−0.200455	−0.100227	+	0.994965i	$0.531957\pi$
$102$	174.004i	0.00165599i
$103$	− 73661.4i	− 0.684143i	−0.939674	−	0.342071i	$-0.888872\pi$
$103$	− 73661.4i	− 0.684143i	0.939674	−	0.342071i	$-0.111128\pi$
$104$	−4279.17	−0.0387950
$105$	0	0
$106$	1046.72	0.00904831
$107$	− 141079.i	− 1.19125i	−0.803262	−	0.595626i	$-0.796904\pi$
$107$	− 141079.i	− 1.19125i	0.803262	−	0.595626i	$-0.203096\pi$
$108$	− 129533.i	− 1.06861i
$109$	3329.82	0.0268444	0.0134222	−	0.999910i	$-0.495727\pi$
$109$	3329.82	0.0268444	0.0134222	+	0.999910i	$0.495727\pi$
$110$	0	0
$111$	19727.3	0.151971
$112$	− 8118.76i	− 0.0611568i
$113$	80229.6i	0.591070i	0.955332	+	0.295535i	$0.0954978\pi$
$113$	80229.6i	0.591070i	−0.955332	+	0.295535i	$0.904502\pi$
$114$	5375.54	0.0387400
$115$	0	0
$116$	213519.	1.47330
$117$	− 18727.0i	− 0.126475i
$118$	1965.91i	0.0129974i
$119$	−307.065	−0.00198776
$120$	0	0
$121$	−48220.5	−0.299411
$122$	17366.4i	0.105636i
$123$	115858.i	0.690499i
$124$	250250.	1.46157
$125$	0	0
$126$	−353.650	−0.00198448
$127$	91691.7i	0.504453i	0.967668	+	0.252227i	$0.0811628\pi$
$127$	91691.7i	0.504453i	−0.967668	+	0.252227i	$0.918837\pi$
$128$	− 51347.2i	− 0.277008i
$129$	124427.	0.658327
$130$	0	0
$131$	−150911.	−0.768319	−0.384160	−	0.923267i	$-0.625509\pi$
$131$	−150911.	−0.768319	−0.384160	+	0.923267i	$0.625509\pi$
$132$	− 122976.i	− 0.614309i
$133$	9486.23i	0.0465013i
$134$	3637.25	0.0174989
$135$	0	0
$136$	−966.211	−0.00447945
$137$	− 20446.3i	− 0.0930710i	−0.998917	−	0.0465355i	$-0.985182\pi$
$137$	− 20446.3i	− 0.0930710i	0.998917	−	0.0465355i	$-0.0148181\pi$
$138$	− 7657.72i	− 0.0342296i
$139$	142881.	0.627245	0.313623	−	0.949548i	$-0.398457\pi$
$139$	142881.	0.627245	0.313623	+	0.949548i	$0.398457\pi$
$140$	0	0
$141$	−149456.	−0.633092
$142$	5168.00i	0.0215081i
$143$	− 56767.5i	− 0.232145i
$144$	111800.	0.449297
$145$	0	0
$146$	−16549.2	−0.0642530
$147$	− 192492.i	− 0.734714i
$148$	54636.0i	0.205033i
$149$	175152.	0.646321	0.323161	−	0.946344i	$-0.395255\pi$
$149$	175152.	0.646321	0.323161	+	0.946344i	$0.395255\pi$
$150$	0	0
$151$	−3588.76	−0.0128086	−0.00640431	−	0.999979i	$-0.502039\pi$
$151$	−3588.76	−0.0128086	−0.00640431	+	0.999979i	$0.502039\pi$
$152$	29849.4i	0.104792i
$153$	− 4228.45i	− 0.0146034i
$154$	−1072.03	−0.00364253
$155$	0	0
$156$	−61872.1	−0.203556
$157$	− 232402.i	− 0.752473i	−0.926524	−	0.376236i	$-0.877218\pi$
$157$	− 232402.i	− 0.752473i	0.926524	−	0.376236i	$-0.122782\pi$
$158$	− 19754.9i	− 0.0629554i
$159$	30343.7	0.0951867
$160$	0	0
$161$	13513.6	0.0410872
$162$	7869.89i	0.0235603i
$163$	144816.i	0.426921i	0.976952	+	0.213460i	$0.0684735\pi$
$163$	144816.i	0.426921i	−0.976952	+	0.213460i	$0.931527\pi$
$164$	−320876.	−0.931596
$165$	0	0
$166$	−34186.3	−0.0962903
$167$	413472.i	1.14724i	0.819121	+	0.573621i	$0.194462\pi$
$167$	413472.i	1.14724i	−0.819121	+	0.573621i	$0.805538\pi$
$168$	2342.62i	0.00640366i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	−130630.	−0.341628
$172$	344610.i	0.888191i
$173$	686615.i	1.74421i	0.489322	+	0.872103i	$0.337244\pi$
$173$	686615.i	1.74421i	−0.489322	+	0.872103i	$0.662756\pi$
$174$	−30576.3	−0.0765618
$175$	0	0
$176$	338900.	0.824689
$177$	56990.2i	0.136731i
$178$	− 26686.6i	− 0.0631312i
$179$	−217412.	−0.507168	−0.253584	−	0.967313i	$-0.581609\pi$
$179$	−217412.	−0.507168	−0.253584	+	0.967313i	$0.581609\pi$
$180$	0	0
$181$	−282439.	−0.640807	−0.320404	−	0.947281i	$-0.603819\pi$
$181$	−282439.	−0.640807	−0.320404	+	0.947281i	$0.603819\pi$
$182$	539.360i	0.00120698i
$183$	503439.i	1.11127i
$184$	42521.9	0.0925908
$185$	0	0
$186$	−35836.4	−0.0759524
$187$	− 12817.8i	− 0.0268046i
$188$	− 413929.i	− 0.854145i
$189$	−32734.1	−0.0666570
$190$	0	0
$191$	−141669.	−0.280990	−0.140495	−	0.990081i	$-0.544869\pi$
$191$	−141669.	−0.280990	−0.140495	+	0.990081i	$0.544869\pi$
$192$	− 365680.i	− 0.715892i
$193$	− 817556.i	− 1.57988i	−0.613183	−	0.789941i	$-0.710111\pi$
$193$	− 817556.i	− 1.57988i	0.613183	−	0.789941i	$-0.289889\pi$
$194$	−69619.4	−0.132808
$195$	0	0
$196$	533118.	0.991251
$197$	566426.i	1.03987i	0.854207	+	0.519933i	$0.174043\pi$
$197$	566426.i	1.03987i	−0.854207	+	0.519933i	$0.825957\pi$
$198$	− 14762.4i	− 0.0267604i
$199$	−400781.	−0.717422	−0.358711	−	0.933449i	$-0.616784\pi$
$199$	−400781.	−0.717422	−0.358711	+	0.933449i	$0.616784\pi$
$200$	0	0
$201$	105441.	0.184086
$202$	8150.43i	0.0140541i
$203$	− 53958.1i	− 0.0919002i
$204$	−13970.4	−0.0235035
$205$	0	0
$206$	−29214.7	−0.0479659
$207$	186089.i	0.301853i
$208$	− 170508.i	− 0.273267i
$209$	−395982.	−0.627062
$210$	0	0
$211$	389440.	0.602191	0.301095	−	0.953594i	$-0.402648\pi$
$211$	389440.	0.602191	0.301095	+	0.953594i	$0.402648\pi$
$212$	84039.0i	0.128423i
$213$	149816.i	0.226261i
$214$	−55953.1	−0.0835198
$215$	0	0
$216$	−103001.	−0.150213
$217$	− 63240.6i	− 0.0911688i
$218$	− 1320.63i	− 0.00188209i
$219$	−479748.	−0.675931
$220$	0	0
$221$	−6448.91	−0.00888190
$222$	− 7823.99i	− 0.0106548i
$223$	− 312613.i	− 0.420964i	−0.977598	−	0.210482i	$-0.932497\pi$
$223$	− 312613.i	− 0.420964i	0.977598	−	0.210482i	$-0.0675033\pi$
$224$	−9740.05	−0.0129700
$225$	0	0
$226$	31819.7	0.0414405
$227$	863698.i	1.11249i	0.831017	+	0.556246i	$0.187759\pi$
$227$	863698.i	1.11249i	−0.831017	+	0.556246i	$0.812241\pi$
$228$	431589.i	0.549837i
$229$	255333.	0.321749	0.160875	−	0.986975i	$-0.448569\pi$
$229$	255333.	0.321749	0.160875	+	0.986975i	$0.448569\pi$
$230$	0	0
$231$	−31077.2	−0.0383189
$232$	− 169785.i	− 0.207099i
$233$	566258.i	0.683321i	0.939823	+	0.341660i	$0.110989\pi$
$233$	566258.i	0.683321i	−0.939823	+	0.341660i	$0.889011\pi$
$234$	−7427.27	−0.00886727
$235$	0	0
$236$	−157838.	−0.184473
$237$	− 572681.i	− 0.662280i
$238$	121.784i	0 0.000139364i
$239$	−1.70328e6	−1.92882	−0.964410	−	0.264412i	$-0.914822\pi$
$239$	−1.70328e6	−1.92882	−0.964410	+	0.264412i	$0.914822\pi$
$240$	0	0
$241$	189220.	0.209857	0.104929	−	0.994480i	$-0.466539\pi$
$241$	189220.	0.209857	0.104929	+	0.994480i	$0.466539\pi$
$242$	19124.6i	0.0209920i
$243$	− 760355.i	− 0.826039i
$244$	−1.39431e6	−1.49928
$245$	0	0
$246$	45950.1	0.0484116
$247$	199228.i	0.207782i
$248$	− 198993.i	− 0.205451i
$249$	−991037.	−1.01296
$250$	0	0
$251$	807394.	0.808911	0.404456	−	0.914558i	$-0.367461\pi$
$251$	807394.	0.808911	0.404456	+	0.914558i	$0.367461\pi$
$252$	− 28393.7i	− 0.0281657i
$253$	564097.i	0.554054i
$254$	36365.6	0.0353677
$255$	0	0
$256$	997412.	0.951206
$257$	1.71583e6i	1.62047i	0.586107	+	0.810234i	$0.300660\pi$
$257$	1.71583e6i	1.62047i	−0.586107	+	0.810234i	$0.699340\pi$
$258$	− 49348.8i	− 0.0461559i
$259$	13807.0	0.0127894
$260$	0	0
$261$	743032.	0.675159
$262$	59852.3i	0.0538676i
$263$	2.09428e6i	1.86701i	0.358568	+	0.933504i	$0.383265\pi$
$263$	2.09428e6i	1.86701i	−0.358568	+	0.933504i	$0.616735\pi$
$264$	−97787.6	−0.0863523
$265$	0	0
$266$	3762.31	0.00326025
$267$	− 773626.i	− 0.664129i
$268$	292026.i	0.248362i
$269$	431530.	0.363605	0.181803	−	0.983335i	$-0.441807\pi$
$269$	431530.	0.363605	0.181803	+	0.983335i	$0.441807\pi$
$270$	0	0
$271$	−290063.	−0.239921	−0.119961	−	0.992779i	$-0.538277\pi$
$271$	−290063.	−0.239921	−0.119961	+	0.992779i	$0.538277\pi$
$272$	− 38499.8i	− 0.0315527i
$273$	15635.7i	0.0126972i
$274$	−8109.18	−0.00652530
$275$	0	0
$276$	614820.	0.485820
$277$	1.89373e6i	1.48292i	0.670995	+	0.741462i	$0.265867\pi$
$277$	1.89373e6i	1.48292i	−0.670995	+	0.741462i	$0.734133\pi$
$278$	− 56667.7i	− 0.0439768i
$279$	870856.	0.669786
$280$	0	0
$281$	−655309.	−0.495086	−0.247543	−	0.968877i	$-0.579623\pi$
$281$	−655309.	−0.495086	−0.247543	+	0.968877i	$0.579623\pi$
$282$	59275.6i	0.0443867i
$283$	− 695619.i	− 0.516304i	−0.966104	−	0.258152i	$-0.916887\pi$
$283$	− 695619.i	− 0.516304i	0.966104	−	0.258152i	$-0.0831135\pi$
$284$	−414926.	−0.305264
$285$	0	0
$286$	−22514.4	−0.0162759
$287$	81088.4i	0.0581104i
$288$	− 134126.i	− 0.0952863i
$289$	1.41840e6	0.998974
$290$	0	0
$291$	−2.01821e6	−1.39712
$292$	− 1.32869e6i	− 0.911943i
$293$	2.44727e6i	1.66538i	0.553742	+	0.832689i	$0.313200\pi$
$293$	2.44727e6i	1.66538i	−0.553742	+	0.832689i	$0.686800\pi$
$294$	−76343.7	−0.0515116
$295$	0	0
$296$	43445.2	0.0288212
$297$	− 1.36641e6i	− 0.898858i
$298$	− 69466.5i	− 0.0453142i
$299$	283809.	0.183590
$300$	0	0
$301$	87086.1	0.0554029
$302$	1423.33i	0 0.000898025i
$303$	236275.i	0.147847i
$304$	−1.18938e6	−0.738138
$305$	0	0
$306$	−1677.04	−0.00102386
$307$	1.41799e6i	0.858674i	0.903144	+	0.429337i	$0.141253\pi$
$307$	1.41799e6i	0.858674i	−0.903144	+	0.429337i	$0.858747\pi$
$308$	− 86070.5i	− 0.0516984i
$309$	−846911.	−0.504594
$310$	0	0
$311$	−146871.	−0.0861062	−0.0430531	−	0.999073i	$-0.513708\pi$
$311$	−146871.	−0.0861062	−0.0430531	+	0.999073i	$0.513708\pi$
$312$	49199.1i	0.0286135i
$313$	2.60983e6i	1.50575i	0.658165	+	0.752873i	$0.271333\pi$
$313$	2.60983e6i	1.50575i	−0.658165	+	0.752873i	$0.728667\pi$
$314$	−92172.4	−0.0527566
$315$	0	0
$316$	1.58608e6	0.893525
$317$	2.85498e6i	1.59571i	0.602847	+	0.797857i	$0.294033\pi$
$317$	2.85498e6i	1.59571i	−0.602847	+	0.797857i	$0.705967\pi$
$318$	− 12034.6i	− 0.00667364i
$319$	2.25237e6	1.23926
$320$	0	0
$321$	−1.62204e6	−0.878615
$322$	− 5359.60i	− 0.00288066i
$323$	44984.4i	0.0239914i
$324$	−631855.	−0.334391
$325$	0	0
$326$	57435.1	0.0299319
$327$	− 38284.1i	− 0.0197993i
$328$	255153.i	0.130953i
$329$	−104604.	−0.0532792
$330$	0	0
$331$	−2.25339e6	−1.13049	−0.565245	−	0.824923i	$-0.691218\pi$
$331$	−2.25339e6	−1.13049	−0.565245	+	0.824923i	$0.691218\pi$
$332$	− 2.74474e6i	− 1.36665i
$333$	190130.i	0.0939594i
$334$	163986.	0.0804342
$335$	0	0
$336$	−93344.3	−0.0451066
$337$	− 795896.i	− 0.381752i	−0.981614	−	0.190876i	$-0.938867\pi$
$337$	− 795896.i	− 0.381752i	0.981614	−	0.190876i	$-0.0611329\pi$
$338$	11327.5i	0.00539315i
$339$	922429.	0.435947
$340$	0	0
$341$	2.63984e6	1.22940
$342$	51809.0i	0.0239519i
$343$	− 269969.i	− 0.123902i
$344$	274025.	0.124852
$345$	0	0
$346$	272317.	0.122288
$347$	− 869632.i	− 0.387714i	−0.981030	−	0.193857i	$-0.937900\pi$
$347$	− 869632.i	− 0.387714i	0.981030	−	0.193857i	$-0.0620998\pi$
$348$	− 2.45490e6i	− 1.08664i
$349$	268145.	0.117844	0.0589219	−	0.998263i	$-0.481234\pi$
$349$	268145.	0.117844	0.0589219	+	0.998263i	$0.481234\pi$
$350$	0	0
$351$	−687473.	−0.297843
$352$	− 406578.i	− 0.174899i
$353$	− 2.22293e6i	− 0.949485i	−0.880125	−	0.474743i	$-0.842541\pi$
$353$	− 2.22293e6i	− 0.949485i	0.880125	−	0.474743i	$-0.157459\pi$
$354$	22602.7	0.00958634
$355$	0	0
$356$	2.14261e6	0.896020
$357$	3530.44i	0.00146608i
$358$	86227.4i	0.0355581i
$359$	−356227.	−0.145878	−0.0729391	−	0.997336i	$-0.523238\pi$
$359$	−356227.	−0.145878	−0.0729391	+	0.997336i	$0.523238\pi$
$360$	0	0
$361$	−1.08639e6	−0.438749
$362$	112017.i	0.0449276i
$363$	554409.i	0.220833i
$364$	−43304.0	−0.0171307
$365$	0	0
$366$	199668.	0.0779122
$367$	1.68973e6i	0.654867i	0.944874	+	0.327433i	$0.106184\pi$
$367$	1.68973e6i	0.654867i	−0.944874	+	0.327433i	$0.893816\pi$
$368$	1.69433e6i	0.652198i
$369$	−1.11663e6	−0.426917
$370$	0	0
$371$	21237.4	0.00801064
$372$	− 2.87722e6i	− 1.07799i
$373$	− 2.01242e6i	− 0.748938i	−0.927239	−	0.374469i	$-0.877825\pi$
$373$	− 2.01242e6i	− 0.748938i	0.927239	−	0.374469i	$-0.122175\pi$
$374$	−5083.63	−0.00187930
$375$	0	0
$376$	−329146.	−0.120066
$377$	− 1.13322e6i	− 0.410638i
$378$	12982.6i	0.00467339i
$379$	3.07308e6	1.09894	0.549472	−	0.835512i	$-0.314829\pi$
$379$	3.07308e6	1.09894	0.549472	+	0.835512i	$0.314829\pi$
$380$	0	0
$381$	1.05421e6	0.372063
$382$	56186.9i	0.0197005i
$383$	− 5.65657e6i	− 1.97041i	−0.171387	−	0.985204i	$-0.554825\pi$
$383$	− 5.65657e6i	− 1.97041i	0.171387	−	0.985204i	$-0.445175\pi$
$384$	−590358.	−0.204309
$385$	0	0
$386$	−324249.	−0.110767
$387$	1.19922e6i	0.407026i
$388$	− 5.58958e6i	− 1.88495i
$389$	1.20141e6	0.402547	0.201274	−	0.979535i	$-0.435492\pi$
$389$	1.20141e6	0.402547	0.201274	+	0.979535i	$0.435492\pi$
$390$	0	0
$391$	64082.6	0.0211981
$392$	− 423922.i	− 0.139338i
$393$	1.73507e6i	0.566678i
$394$	224649.	0.0729060
$395$	0	0
$396$	1.18524e6	0.379810
$397$	− 1.91438e6i	− 0.609610i	−0.952415	−	0.304805i	$-0.901409\pi$
$397$	− 1.91438e6i	− 0.609610i	0.952415	−	0.304805i	$-0.0985912\pi$
$398$	158953.i	0.0502992i
$399$	109067.	0.0342973
$400$	0	0
$401$	−5.02610e6	−1.56088	−0.780441	−	0.625229i	$-0.785006\pi$
$401$	−5.02610e6	−1.56088	−0.780441	+	0.625229i	$0.785006\pi$
$402$	− 41818.8i	− 0.0129064i
$403$	− 1.32816e6i	− 0.407370i
$404$	−654379.	−0.199469
$405$	0	0
$406$	−21400.2	−0.00644322
$407$	576345.i	0.172463i
$408$	11108.9i	0.00330385i
$409$	2.26919e6	0.670754	0.335377	−	0.942084i	$-0.391136\pi$
$409$	2.26919e6	0.670754	0.335377	+	0.942084i	$0.391136\pi$
$410$	0	0
$411$	−235079.	−0.0686450
$412$	− 2.34558e6i	− 0.680780i
$413$	39887.1i	0.0115069i
$414$	73804.5	0.0211632
$415$	0	0
$416$	−204558.	−0.0579540
$417$	− 1.64275e6i	− 0.462629i
$418$	157050.i	0.0439639i
$419$	3.92074e6	1.09102	0.545510	−	0.838104i	$-0.316336\pi$
$419$	3.92074e6	1.09102	0.545510	+	0.838104i	$0.316336\pi$
$420$	0	0
$421$	1.77811e6	0.488939	0.244469	−	0.969657i	$-0.421386\pi$
$421$	1.77811e6	0.488939	0.244469	+	0.969657i	$0.421386\pi$
$422$	− 154455.i	− 0.0422202i
$423$	− 1.44045e6i	− 0.391424i
$424$	66825.7	0.0180521
$425$	0	0
$426$	59418.3	0.0158634
$427$	352355.i	0.0935212i
$428$	− 4.49234e6i	− 1.18540i
$429$	−652677.	−0.171220
$430$	0	0
$431$	−4.45663e6	−1.15561	−0.577807	−	0.816173i	$-0.696091\pi$
$431$	−4.45663e6	−1.15561	−0.577807	+	0.816173i	$0.696091\pi$
$432$	− 4.10419e6i	− 1.05808i
$433$	1.25271e6i	0.321093i	0.987028	+	0.160547i	$0.0513257\pi$
$433$	1.25271e6i	0.321093i	−0.987028	+	0.160547i	$0.948674\pi$
$434$	−25081.7	−0.00639194
$435$	0	0
$436$	106030.	0.0267125
$437$	− 1.97972e6i	− 0.495906i
$438$	190272.i	0.0473902i
$439$	−3.75453e6	−0.929810	−0.464905	−	0.885360i	$-0.653912\pi$
$439$	−3.75453e6	−0.929810	−0.464905	+	0.885360i	$0.653912\pi$
$440$	0	0
$441$	1.85522e6	0.454254
$442$	2557.69i	0 0.000622719i
$443$	2.31259e6i	0.559874i	0.960018	+	0.279937i	$0.0903135\pi$
$443$	2.31259e6i	0.559874i	−0.960018	+	0.279937i	$0.909686\pi$
$444$	628170.	0.151224
$445$	0	0
$446$	−123985.	−0.0295142
$447$	− 2.01378e6i	− 0.476698i
$448$	− 255937.i	− 0.0602474i
$449$	7.80175e6	1.82632	0.913159	−	0.407603i	$-0.133635\pi$
$449$	7.80175e6	1.82632	0.913159	+	0.407603i	$0.133635\pi$
$450$	0	0
$451$	−3.38486e6	−0.783609
$452$	2.55473e6i	0.588164i
$453$	41261.3i	0.00944708i
$454$	342549.	0.0779980
$455$	0	0
$456$	343189.	0.0772896
$457$	4.88928e6i	1.09510i	0.836772	+	0.547551i	$0.184440\pi$
$457$	4.88928e6i	1.09510i	−0.836772	+	0.547551i	$0.815560\pi$
$458$	− 101267.i	− 0.0225582i
$459$	−155228.	−0.0343904
$460$	0	0
$461$	−4.08764e6	−0.895818	−0.447909	−	0.894079i	$-0.647831\pi$
$461$	−4.08764e6	−0.895818	−0.447909	+	0.894079i	$0.647831\pi$
$462$	12325.5i	0.00268657i
$463$	− 3.13129e6i	− 0.678845i	−0.940634	−	0.339423i	$-0.889768\pi$
$463$	− 3.13129e6i	− 0.678845i	0.940634	−	0.339423i	$-0.110232\pi$
$464$	6.76526e6	1.45878
$465$	0	0
$466$	224582.	0.0479083
$467$	5.78075e6i	1.22657i	0.789862	+	0.613284i	$0.210152\pi$
$467$	5.78075e6i	1.22657i	−0.789862	+	0.613284i	$0.789848\pi$
$468$	− 596318.i	− 0.125853i
$469$	73797.7	0.0154921
$470$	0	0
$471$	−2.67201e6	−0.554991
$472$	125509.i	0.0259310i
$473$	3.63522e6i	0.747099i
$474$	−227130.	−0.0464331
$475$	0	0
$476$	−9777.79	−0.00197799
$477$	292451.i	0.0588514i
$478$	675535.i	0.135231i
$479$	5.98034e6	1.19093	0.595467	−	0.803380i	$-0.296967\pi$
$479$	5.98034e6	1.19093	0.595467	+	0.803380i	$0.296967\pi$
$480$	0	0
$481$	289972.	0.0571469
$482$	− 75045.9i	− 0.0147133i
$483$	− 155371.i	− 0.0303041i
$484$	−1.53547e6	−0.297940
$485$	0	0
$486$	−301563.	−0.0579144
$487$	− 8.62843e6i	− 1.64858i	−0.566169	−	0.824289i	$-0.691575\pi$
$487$	− 8.62843e6i	− 1.64858i	0.566169	−	0.824289i	$-0.308425\pi$
$488$	1.10872e6i	0.210752i
$489$	1.66500e6	0.314878
$490$	0	0
$491$	−4.67336e6	−0.874834	−0.437417	−	0.899259i	$-0.644107\pi$
$491$	−4.67336e6	−0.874834	−0.437417	+	0.899259i	$0.644107\pi$
$492$	3.68923e6i	0.687105i
$493$	− 255873.i	− 0.0474142i
$494$	79015.2	0.0145678
$495$	0	0
$496$	7.92909e6	1.44717
$497$	104856.i	0.0190415i
$498$	393053.i	0.0710195i
$499$	−1.02014e7	−1.83404	−0.917018	−	0.398847i	$-0.869411\pi$
$499$	−1.02014e7	−1.83404	−0.917018	+	0.398847i	$0.869411\pi$
$500$	0	0
$501$	4.75383e6	0.846155
$502$	− 320218.i	− 0.0567136i
$503$	− 1.14659e6i	− 0.202064i	−0.994883	−	0.101032i	$-0.967786\pi$
$503$	− 1.14659e6i	− 0.202064i	0.994883	−	0.101032i	$-0.0322144\pi$
$504$	−22578.0	−0.00395921
$505$	0	0
$506$	223725.	0.0388453
$507$	328376.i	0.0567351i
$508$	2.91971e6i	0.501974i
$509$	2.19167e6	0.374957	0.187478	−	0.982269i	$-0.439969\pi$
$509$	2.19167e6	0.374957	0.187478	+	0.982269i	$0.439969\pi$
$510$	0	0
$511$	−335773.	−0.0568844
$512$	− 2.03869e6i	− 0.343698i
$513$	4.79548e6i	0.804523i
$514$	680509.	0.113613
$515$	0	0
$516$	3.96210e6	0.655091
$517$	− 4.36646e6i	− 0.718461i
$518$	− 5475.97i	0 0.000896679i
$519$	7.89426e6	1.28645
$520$	0	0
$521$	−6.90308e6	−1.11416	−0.557082	−	0.830458i	$-0.688079\pi$
$521$	−6.90308e6	−1.11416	−0.557082	+	0.830458i	$0.688079\pi$
$522$	− 294692.i	− 0.0473361i
$523$	1.59809e6i	0.255473i	0.991808	+	0.127737i	$0.0407713\pi$
$523$	1.59809e6i	0.255473i	−0.991808	+	0.127737i	$0.959229\pi$
$524$	−4.80540e6	−0.764542
$525$	0	0
$526$	830609.	0.130898
$527$	− 299892.i	− 0.0470368i
$528$	− 3.89646e6i	− 0.608254i
$529$	3.61614e6	0.561831
$530$	0	0
$531$	−549267.	−0.0845371
$532$	302067.i	0.0462727i
$533$	1.70300e6i	0.259655i
$534$	−306826.	−0.0465628
$535$	0	0
$536$	232212.	0.0349118
$537$	2.49967e6i	0.374065i
$538$	− 171148.i	− 0.0254927i
$539$	5.62376e6	0.833787
$540$	0	0
$541$	−279387.	−0.0410405	−0.0205202	−	0.999789i	$-0.506532\pi$
$541$	−279387.	−0.0410405	−0.0205202	+	0.999789i	$0.506532\pi$
$542$	115041.i	0.0168211i
$543$	3.24730e6i	0.472631i
$544$	−46188.1	−0.00669164
$545$	0	0
$546$	6201.22	0.000890216 0
$547$	606222.i	0.0866290i	0.999061	+	0.0433145i	$0.0137918\pi$
$547$	606222.i	0.0866290i	−0.999061	+	0.0433145i	$0.986208\pi$
$548$	− 651067.i	− 0.0926135i
$549$	−4.85211e6	−0.687068
$550$	0	0
$551$	−7.90475e6	−1.10920
$552$	− 488890.i	− 0.0682909i
$553$	− 400816.i	− 0.0557356i
$554$	751068.	0.103969
$555$	0	0
$556$	4.54972e6	0.624162
$557$	− 3.37279e6i	− 0.460629i	−0.973116	−	0.230315i	$-0.926024\pi$
$557$	− 3.37279e6i	− 0.460629i	0.973116	−	0.230315i	$-0.0739755\pi$
$558$	− 345388.i	− 0.0469593i
$559$	1.82896e6	0.247557
$560$	0	0
$561$	−147371.	−0.0197699
$562$	259901.i	0.0347110i
$563$	3.75518e6i	0.499298i	0.968336	+	0.249649i	$0.0803153\pi$
$563$	3.75518e6i	0.499298i	−0.968336	+	0.249649i	$0.919685\pi$
$564$	−4.75910e6	−0.629980
$565$	0	0
$566$	−275888.	−0.0361986
$567$	159676.i	0.0208584i
$568$	329939.i	0.0429104i
$569$	−5.11904e6	−0.662839	−0.331420	−	0.943483i	$-0.607528\pi$
$569$	−5.11904e6	−0.662839	−0.331420	+	0.943483i	$0.607528\pi$
$570$	0	0
$571$	1.23612e7	1.58662	0.793308	−	0.608821i	$-0.208357\pi$
$571$	1.23612e7	1.58662	0.793308	+	0.608821i	$0.208357\pi$
$572$	− 1.80763e6i	− 0.231004i
$573$	1.62882e6i	0.207246i
$574$	32160.3	0.00407418
$575$	0	0
$576$	3.52439e6	0.442617
$577$	1.46773e6i	0.183530i	0.995781	+	0.0917651i	$0.0292509\pi$
$577$	1.46773e6i	0.183530i	−0.995781	+	0.0917651i	$0.970749\pi$
$578$	− 562549.i	− 0.0700391i
$579$	−9.39974e6	−1.16525
$580$	0	0
$581$	−693622.	−0.0852476
$582$	800439.i	0.0979537i
$583$	886511.i	0.108022i
$584$	−1.05654e6	−0.128190
$585$	0	0
$586$	970606.	0.116761
$587$	6.01770e6i	0.720834i	0.932791	+	0.360417i	$0.117366\pi$
$587$	6.01770e6i	0.720834i	−0.932791	+	0.360417i	$0.882634\pi$
$588$	− 6.12945e6i	− 0.731103i
$589$	−9.26461e6	−1.10037
$590$	0	0
$591$	6.51240e6	0.766959
$592$	1.73112e6i	0.203013i
$593$	1.28954e7i	1.50590i	0.658076	+	0.752951i	$0.271371\pi$
$593$	1.28954e7i	1.50590i	−0.658076	+	0.752951i	$0.728629\pi$
$594$	−541930.	−0.0630199
$595$	0	0
$596$	5.57730e6	0.643144
$597$	4.60793e6i	0.529139i
$598$	− 112561.i	− 0.0128717i
$599$	−1.07891e7	−1.22863	−0.614313	−	0.789062i	$-0.710567\pi$
$599$	−1.07891e7	−1.22863	−0.614313	+	0.789062i	$0.710567\pi$
$600$	0	0
$601$	3.51995e6	0.397512	0.198756	−	0.980049i	$-0.436310\pi$
$601$	3.51995e6	0.397512	0.198756	+	0.980049i	$0.436310\pi$
$602$	− 34539.0i	− 0.00388435i
$603$	1.01623e6i	0.113815i
$604$	−114276.	−0.0127457
$605$	0	0
$606$	93708.5	0.0103657
$607$	− 1.43143e6i	− 0.157688i	−0.996887	−	0.0788439i	$-0.974877\pi$
$607$	− 1.43143e6i	− 0.157688i	0.996887	−	0.0788439i	$-0.0251229\pi$
$608$	1.42690e6i	0.156543i
$609$	−620376.	−0.0677816
$610$	0	0
$611$	−2.19686e6	−0.238067
$612$	− 134645.i	− 0.0145316i
$613$	− 1.06458e7i	− 1.14426i	−0.820162	−	0.572131i	$-0.806117\pi$
$613$	− 1.06458e7i	− 1.14426i	0.820162	−	0.572131i	$-0.193883\pi$
$614$	562387.	0.0602025
$615$	0	0
$616$	−68441.1	−0.00726716
$617$	9.96269e6i	1.05357i	0.849998	+	0.526785i	$0.176603\pi$
$617$	9.96269e6i	1.05357i	−0.849998	+	0.526785i	$0.823397\pi$
$618$	335892.i	0.0353776i
$619$	−1.14066e7	−1.19655	−0.598273	−	0.801292i	$-0.704146\pi$
$619$	−1.14066e7	−1.19655	−0.598273	+	0.801292i	$0.704146\pi$
$620$	0	0
$621$	6.83139e6	0.710854
$622$	58250.1i	0.00603699i
$623$	− 541457.i	− 0.0558912i
$624$	−1.96039e6	−0.201550
$625$	0	0
$626$	1.03508e6	0.105569
$627$	4.55275e6i	0.462493i
$628$	− 7.40031e6i	− 0.748774i
$629$	65474.0	0.00659845
$630$	0	0
$631$	−1.56307e7	−1.56280	−0.781402	−	0.624028i	$-0.785495\pi$
$631$	−1.56307e7	−1.56280	−0.781402	+	0.624028i	$0.785495\pi$
$632$	− 1.26121e6i	− 0.125601i
$633$	− 4.47753e6i	− 0.444150i
$634$	1.13231e6	0.111877
$635$	0	0
$636$	966227.	0.0947188
$637$	− 2.82944e6i	− 0.276282i
$638$	− 893306.i	− 0.0868857i
$639$	−1.44392e6	−0.139891
$640$	0	0
$641$	4.75966e6	0.457542	0.228771	−	0.973480i	$-0.426529\pi$
$641$	4.75966e6	0.457542	0.228771	+	0.973480i	$0.426529\pi$
$642$	643313.i	0.0616006i
$643$	946165.i	0.0902484i	0.998981	+	0.0451242i	$0.0143684\pi$
$643$	946165.i	0.0902484i	−0.998981	+	0.0451242i	$0.985632\pi$
$644$	430310.	0.0408852
$645$	0	0
$646$	17841.2	0.00168206
$647$	870173.i	0.0817231i	0.999165	+	0.0408616i	$0.0130103\pi$
$647$	870173.i	0.0817231i	−0.999165	+	0.0408616i	$0.986990\pi$
$648$	502435.i	0.0470048i
$649$	−1.66500e6	−0.155168
$650$	0	0
$651$	−727100.	−0.0672421
$652$	4.61133e6i	0.424822i
$653$	− 2.57122e6i	− 0.235970i	−0.993015	−	0.117985i	$-0.962357\pi$
$653$	− 2.57122e6i	− 0.235970i	0.993015	−	0.117985i	$-0.0376434\pi$
$654$	−15183.8	−0.00138815
$655$	0	0
$656$	−1.01668e7	−0.922415
$657$	− 4.62377e6i	− 0.417910i
$658$	41486.7i	0.00373546i
$659$	−1.09142e7	−0.978988	−0.489494	−	0.872007i	$-0.662818\pi$
$659$	−1.09142e7	−0.978988	−0.489494	+	0.872007i	$0.662818\pi$
$660$	0	0
$661$	4.31733e6	0.384337	0.192168	−	0.981362i	$-0.438448\pi$
$661$	4.31733e6	0.384337	0.192168	+	0.981362i	$0.438448\pi$
$662$	893712.i	0.0792597i
$663$	74145.5i	0.00655090i
$664$	−2.18255e6	−0.192107
$665$	0	0
$666$	75407.0	0.00658758
$667$	1.12607e7i	0.980057i
$668$	1.31661e7i	1.14160i
$669$	−3.59422e6	−0.310484
$670$	0	0
$671$	−1.47083e7	−1.26112
$672$	111985.i	0.00956613i
$673$	2.02793e7i	1.72590i	0.505292	+	0.862949i	$0.331385\pi$
$673$	2.02793e7i	1.72590i	−0.505292	+	0.862949i	$0.668615\pi$
$674$	−315658.	−0.0267650
$675$	0	0
$676$	−909459.	−0.0765450
$677$	− 1.37469e7i	− 1.15275i	−0.817186	−	0.576373i	$-0.804467\pi$
$677$	− 1.37469e7i	− 1.15275i	0.817186	−	0.576373i	$-0.195533\pi$
$678$	− 365842.i	− 0.0305647i
$679$	−1.41254e6	−0.117578
$680$	0	0
$681$	9.93025e6	0.820526
$682$	− 1.04698e6i	− 0.0861942i
$683$	− 1.37681e7i	− 1.12933i	−0.825320	−	0.564665i	$-0.809005\pi$
$683$	− 1.37681e7i	− 1.12933i	0.825320	−	0.564665i	$-0.190995\pi$
$684$	−4.15963e6	−0.339949
$685$	0	0
$686$	−107072.	−0.00868690
$687$	− 2.93565e6i	− 0.237308i
$688$	1.09188e7i	0.879438i
$689$	446023.	0.0357939
$690$	0	0
$691$	6.60598e6	0.526310	0.263155	−	0.964754i	$-0.415237\pi$
$691$	6.60598e6	0.526310	0.263155	+	0.964754i	$0.415237\pi$
$692$	2.18637e7i	1.73563i
$693$	− 299520.i	− 0.0236915i
$694$	−344903.	−0.0271830
$695$	0	0
$696$	−1.95207e6	−0.152747
$697$	384527.i	0.0299809i
$698$	− 106348.i	− 0.00826214i
$699$	6.51047e6	0.503987
$700$	0	0
$701$	−1.37908e7	−1.05997	−0.529985	−	0.848007i	$-0.677803\pi$
$701$	−1.37908e7	−1.05997	−0.529985	+	0.848007i	$0.677803\pi$
$702$	272657.i	0.0208821i
$703$	− 2.02270e6i	− 0.154363i
$704$	1.06836e7	0.812426
$705$	0	0
$706$	−881630.	−0.0665694
$707$	165368.i	0.0124423i
$708$	1.81472e6i	0.136059i
$709$	−1.96433e7	−1.46757	−0.733785	−	0.679382i	$-0.762248\pi$
$709$	−1.96433e7	−1.46757	−0.733785	+	0.679382i	$0.762248\pi$
$710$	0	0
$711$	5.51945e6	0.409470
$712$	− 1.70375e6i	− 0.125952i
$713$	1.31979e7i	0.972257i
$714$	1400.20	0.000102788 0
$715$	0	0
$716$	−6.92300e6	−0.504675
$717$	1.95832e7i	1.42261i
$718$	141282.i	0.0102277i
$719$	2.26567e7	1.63446	0.817229	−	0.576312i	$-0.195509\pi$
$719$	2.26567e7	1.63446	0.817229	+	0.576312i	$0.195509\pi$
$720$	0	0
$721$	−592749.	−0.0424652
$722$	430869.i	0.0307611i
$723$	− 2.17553e6i	− 0.154781i
$724$	−8.99361e6	−0.637657
$725$	0	0
$726$	219883.	0.0154828
$727$	1.36449e7i	0.957491i	0.877954	+	0.478746i	$0.158908\pi$
$727$	1.36449e7i	0.957491i	−0.877954	+	0.478746i	$0.841092\pi$
$728$	34434.2i	0.00240803i
$729$	−1.35639e7	−0.945293
$730$	0	0
$731$	412969.	0.0285841
$732$	1.60309e7i	1.10581i
$733$	1.07727e7i	0.740569i	0.928918	+	0.370284i	$0.120740\pi$
$733$	1.07727e7i	0.740569i	−0.928918	+	0.370284i	$0.879260\pi$
$734$	670161.	0.0459134
$735$	0	0
$736$	2.03269e6	0.138317
$737$	3.08053e6i	0.208909i
$738$	442864.i	0.0299316i
$739$	−8.30186e6	−0.559196	−0.279598	−	0.960117i	$-0.590201\pi$
$739$	−8.30186e6	−0.559196	−0.279598	+	0.960117i	$0.590201\pi$
$740$	0	0
$741$	2.29059e6	0.153251
$742$	− 8422.93i	0 0.000561634i
$743$	2.21659e7i	1.47304i	0.676417	+	0.736519i	$0.263532\pi$
$743$	2.21659e7i	1.47304i	−0.676417	+	0.736519i	$0.736468\pi$
$744$	−2.28789e6	−0.151531
$745$	0	0
$746$	−798140.	−0.0525088
$747$	− 9.55153e6i	− 0.626284i
$748$	− 408153.i	− 0.0266728i
$749$	−1.13526e6	−0.0739417
$750$	0	0
$751$	2.47886e7	1.60380	0.801902	−	0.597455i	$-0.203821\pi$
$751$	2.47886e7	1.60380	0.801902	+	0.597455i	$0.203821\pi$
$752$	− 1.31152e7i	− 0.845727i
$753$	− 9.28290e6i	− 0.596618i
$754$	−449442.	−0.0287902
$755$	0	0
$756$	−1.04234e6	−0.0663293
$757$	− 1.63378e6i	− 0.103623i	−0.998657	−	0.0518113i	$-0.983501\pi$
$757$	− 1.63378e6i	− 0.103623i	0.998657	−	0.0518113i	$-0.0164994\pi$
$758$	− 1.21881e6i	− 0.0770481i
$759$	6.48562e6	0.408646
$760$	0	0
$761$	−1.29552e7	−0.810931	−0.405465	−	0.914110i	$-0.632891\pi$
$761$	−1.29552e7	−0.810931	−0.405465	+	0.914110i	$0.632891\pi$
$762$	− 418109.i	− 0.0260857i
$763$	− 26794.9i	− 0.00166625i
$764$	−4.51111e6	−0.279608
$765$	0	0
$766$	−2.24344e6	−0.138147
$767$	837700.i	0.0514162i
$768$	− 1.14676e7i	− 0.701568i
$769$	1.14055e7	0.695502	0.347751	−	0.937587i	$-0.386945\pi$
$769$	1.14055e7	0.695502	0.347751	+	0.937587i	$0.386945\pi$
$770$	0	0
$771$	1.97275e7	1.19519
$772$	− 2.60332e7i	− 1.57212i
$773$	6.83570e6i	0.411467i	0.978608	+	0.205733i	$0.0659579\pi$
$773$	6.83570e6i	0.411467i	−0.978608	+	0.205733i	$0.934042\pi$
$774$	475620.	0.0285370
$775$	0	0
$776$	−4.44469e6	−0.264964
$777$	− 158744.i	− 0.00943291i
$778$	− 476488.i	− 0.0282230i
$779$	1.18793e7	0.701369
$780$	0	0
$781$	−4.37698e6	−0.256772
$782$	− 25415.6i	− 0.00148622i
$783$	− 2.72769e7i	− 1.58998i
$784$	1.68917e7	0.981481
$785$	0	0
$786$	688144.	0.0397304
$787$	− 2.71866e7i	− 1.56465i	−0.622868	−	0.782327i	$-0.714033\pi$
$787$	− 2.71866e7i	− 1.56465i	0.622868	−	0.782327i	$-0.285967\pi$
$788$	1.80365e7i	1.03475i
$789$	2.40787e7	1.37702
$790$	0	0
$791$	645603.	0.0366880
$792$	− 942469.i	− 0.0533893i
$793$	7.40006e6i	0.417881i
$794$	−759258.	−0.0427403
$795$	0	0
$796$	−1.27620e7	−0.713896
$797$	1.81329e7i	1.01117i	0.862778	+	0.505583i	$0.168722\pi$
$797$	1.81329e7i	1.01117i	−0.862778	+	0.505583i	$0.831278\pi$
$798$	− 43256.7i	− 0.00240462i
$799$	−496039.	−0.0274884
$800$	0	0
$801$	7.45614e6	0.410613
$802$	1.99339e6i	0.109435i
$803$	− 1.40161e7i	− 0.767077i
$804$	3.35753e6	0.183181
$805$	0	0
$806$	−526760.	−0.0285611
$807$	− 4.96146e6i	− 0.268179i
$808$	520346.i	0.0280391i
$809$	−3.89183e6	−0.209065	−0.104533	−	0.994521i	$-0.533335\pi$
$809$	−3.89183e6	−0.209065	−0.104533	+	0.994521i	$0.533335\pi$
$810$	0	0
$811$	−1.40881e7	−0.752142	−0.376071	−	0.926591i	$-0.622725\pi$
$811$	−1.40881e7	−0.752142	−0.376071	+	0.926591i	$0.622725\pi$
$812$	− 1.71817e6i	− 0.0914485i
$813$	3.33496e6i	0.176955i
$814$	228583.	0.0120916
$815$	0	0
$816$	−442646.	−0.0232719
$817$	− 1.27579e7i	− 0.668691i
$818$	− 899980.i	− 0.0470272i
$819$	−150695.	−0.00785036
$820$	0	0
$821$	3.32912e7	1.72374	0.861871	−	0.507128i	$-0.169293\pi$
$821$	3.32912e7	1.72374	0.861871	+	0.507128i	$0.169293\pi$
$822$	93234.1i	0.00481277i
$823$	− 3.41282e7i	− 1.75636i	−0.478330	−	0.878180i	$-0.658758\pi$
$823$	− 3.41282e7i	− 1.75636i	0.478330	−	0.878180i	$-0.341242\pi$
$824$	−1.86514e6	−0.0956961
$825$	0	0
$826$	15819.5	0.000806759 0
$827$	− 1.68225e7i	− 0.855317i	−0.903940	−	0.427658i	$-0.859339\pi$
$827$	− 1.68225e7i	− 0.855317i	0.903940	−	0.427658i	$-0.140661\pi$
$828$	5.92559e6i	0.300370i
$829$	2.49836e7	1.26261	0.631305	−	0.775534i	$-0.282520\pi$
$829$	2.49836e7	1.26261	0.631305	+	0.775534i	$0.282520\pi$
$830$	0	0
$831$	2.17729e7	1.09374
$832$	− 5.37513e6i	− 0.269204i
$833$	− 638871.i	− 0.0319007i
$834$	−651529.	−0.0324354
$835$	0	0
$836$	−1.26092e7	−0.623979
$837$	− 3.19693e7i	− 1.57732i
$838$	− 1.55500e6i	− 0.0764925i
$839$	−3.58243e7	−1.75700	−0.878501	−	0.477740i	$-0.841456\pi$
$839$	−3.58243e7	−1.75700	−0.878501	+	0.477740i	$0.841456\pi$
$840$	0	0
$841$	2.44514e7	1.19211
$842$	− 705214.i	− 0.0342800i
$843$	7.53433e6i	0.365153i
$844$	1.24008e7	0.599231
$845$	0	0
$846$	−571293.	−0.0274431
$847$	388028.i	0.0185846i
$848$	2.66274e6i	0.127157i
$849$	−7.99778e6	−0.380803
$850$	0	0
$851$	−2.88144e6	−0.136391
$852$	4.77056e6i	0.225149i
$853$	− 1.13433e7i	− 0.533786i	−0.963726	−	0.266893i	$-0.914003\pi$
$853$	− 1.13433e7i	− 0.533786i	0.963726	−	0.266893i	$-0.0859970\pi$
$854$	139747.	0.00655687
$855$	0	0
$856$	−3.57220e6	−0.166629
$857$	2.46384e7i	1.14594i	0.819578	+	0.572968i	$0.194208\pi$
$857$	2.46384e7i	1.14594i	−0.819578	+	0.572968i	$0.805792\pi$
$858$	258857.i	0.0120044i
$859$	−1.82989e6	−0.0846140	−0.0423070	−	0.999105i	$-0.513471\pi$
$859$	−1.82989e6	−0.0846140	−0.0423070	+	0.999105i	$0.513471\pi$
$860$	0	0
$861$	932302.	0.0428597
$862$	1.76753e6i	0.0810213i
$863$	− 2.26407e6i	− 0.103482i	−0.998661	−	0.0517408i	$-0.983523\pi$
$863$	− 2.26407e6i	− 0.103482i	0.998661	−	0.0517408i	$-0.0164770\pi$
$864$	−4.92379e6	−0.224396
$865$	0	0
$866$	496835.	0.0225122
$867$	− 1.63079e7i	− 0.736800i
$868$	− 2.01375e6i	− 0.0907207i
$869$	1.67312e7	0.751585
$870$	0	0
$871$	1.54988e6	0.0692234
$872$	− 84312.7i	− 0.00375493i
$873$	− 1.94514e7i	− 0.863803i
$874$	−785171.	−0.0347685
$875$	0	0
$876$	−1.52765e7	−0.672609
$877$	1.77367e7i	0.778708i	0.921088	+	0.389354i	$0.127302\pi$
$877$	1.77367e7i	0.778708i	−0.921088	+	0.389354i	$0.872698\pi$
$878$	1.48908e6i	0.0651899i
$879$	2.81371e7	1.22831
$880$	0	0
$881$	−3.79053e7	−1.64536	−0.822679	−	0.568506i	$-0.807522\pi$
$881$	−3.79053e7	−1.64536	−0.822679	+	0.568506i	$0.807522\pi$
$882$	− 735794.i	− 0.0318482i
$883$	− 3.29131e7i	− 1.42058i	−0.703908	−	0.710291i	$-0.748563\pi$
$883$	− 3.29131e7i	− 1.42058i	0.703908	−	0.710291i	$-0.251437\pi$
$884$	−205351.	−0.00883824
$885$	0	0
$886$	917192.	0.0392533
$887$	9.54706e6i	0.407437i	0.979029	+	0.203719i	$0.0653027\pi$
$887$	9.54706e6i	0.407437i	−0.979029	+	0.203719i	$0.934697\pi$
$888$	− 499505.i	− 0.0212573i
$889$	737838.	0.0313117
$890$	0	0
$891$	−6.66531e6	−0.281272
$892$	− 9.95444e6i	− 0.418894i
$893$	1.53242e7i	0.643058i
$894$	−798681.	−0.0334218
$895$	0	0
$896$	−413188.	−0.0171940
$897$	− 3.26306e6i	− 0.135408i
$898$	− 3.09424e6i	− 0.128045i
$899$	5.26975e7	2.17466
$900$	0	0
$901$	100710.	0.00413293
$902$	1.34246e6i	0.0549396i
$903$	− 1.00126e6i	− 0.0408627i
$904$	2.03145e6	0.0826773
$905$	0	0
$906$	16364.6	0.000662344 0
$907$	− 2.11768e7i	− 0.854758i	−0.904073	−	0.427379i	$-0.859437\pi$
$907$	− 2.11768e7i	− 0.854758i	0.904073	−	0.427379i	$-0.140563\pi$
$908$	2.75025e7i	1.10702i
$909$	−2.27720e6	−0.0914095
$910$	0	0
$911$	−4.30962e7	−1.72045	−0.860227	−	0.509912i	$-0.829678\pi$
$911$	−4.30962e7	−1.72045	−0.860227	+	0.509912i	$0.829678\pi$
$912$	1.36748e7i	0.544418i
$913$	− 2.89537e7i	− 1.14955i
$914$	1.93913e6	0.0767787
$915$	0	0
$916$	8.13048e6	0.320168
$917$	1.21437e6i	0.0476900i
$918$	61564.4i	0.00241114i
$919$	−1.40263e6	−0.0547841	−0.0273920	−	0.999625i	$-0.508720\pi$
$919$	−1.40263e6	−0.0547841	−0.0273920	+	0.999625i	$0.508720\pi$
$920$	0	0
$921$	1.63032e7	0.633320
$922$	1.62119e6i	0.0628067i
$923$	2.20216e6i	0.0850832i
$924$	−989583.	−0.0381305
$925$	0	0
$926$	−1.24189e6	−0.0475945
$927$	− 8.16247e6i	− 0.311977i
$928$	− 8.11626e6i	− 0.309375i
$929$	−1.27299e7	−0.483934	−0.241967	−	0.970285i	$-0.577793\pi$
$929$	−1.27299e7	−0.483934	−0.241967	+	0.970285i	$0.577793\pi$
$930$	0	0
$931$	−1.97368e7	−0.746281
$932$	1.80312e7i	0.679962i
$933$	1.68863e6i	0.0635081i
$934$	2.29269e6	0.0859960
$935$	0	0
$936$	−474177.	−0.0176909
$937$	− 4.02938e7i	− 1.49930i	−0.661833	−	0.749651i	$-0.730221\pi$
$937$	− 4.02938e7i	− 1.49930i	0.661833	−	0.749651i	$-0.269779\pi$
$938$	− 29268.7i	− 0.00108617i
$939$	3.00062e7	1.11057
$940$	0	0
$941$	−2.47470e7	−0.911064	−0.455532	−	0.890219i	$-0.650551\pi$
$941$	−2.47470e7	−0.911064	−0.455532	+	0.890219i	$0.650551\pi$
$942$	1.05974e6i	0.0389110i
$943$	− 1.69226e7i	− 0.619710i
$944$	−5.00104e6	−0.182655
$945$	0	0
$946$	1.44176e6	0.0523798
$947$	3.28379e7i	1.18987i	0.803772	+	0.594937i	$0.202823\pi$
$947$	3.28379e7i	1.18987i	−0.803772	+	0.594937i	$0.797177\pi$
$948$	− 1.82357e7i	− 0.659025i
$949$	−7.05182e6	−0.254177
$950$	0	0
$951$	3.28248e7	1.17693
$952$	7775.05i	0 0.000278042i
$953$	− 2.12237e7i	− 0.756989i	−0.925604	−	0.378494i	$-0.876442\pi$
$953$	− 2.12237e7i	− 0.756989i	0.925604	−	0.378494i	$-0.123558\pi$
$954$	115988.	0.00412613
$955$	0	0
$956$	−5.42371e7	−1.91934
$957$	− 2.58963e7i	− 0.914023i
$958$	− 2.37185e6i	− 0.0834975i
$959$	−164531.	−0.00577697
$960$	0	0
$961$	3.31340e7	1.15735
$962$	− 115005.i	− 0.00400663i
$963$	− 1.56331e7i	− 0.543224i
$964$	6.02526e6	0.208825
$965$	0	0
$966$	−61621.3	−0.00212465
$967$	− 5.66776e7i	− 1.94915i	−0.224060	−	0.974575i	$-0.571931\pi$
$967$	− 5.66776e7i	− 1.94915i	0.224060	−	0.974575i	$-0.428069\pi$
$968$	1.22097e6i	0.0418809i
$969$	517202.	0.0176950
$970$	0	0
$971$	−3.20888e7	−1.09221	−0.546104	−	0.837718i	$-0.683889\pi$
$971$	−3.20888e7	−1.09221	−0.546104	+	0.837718i	$0.683889\pi$
$972$	− 2.42118e7i	− 0.821979i
$973$	− 1.14976e6i	− 0.0389335i
$974$	−3.42210e6	−0.115583
$975$	0	0
$976$	−4.41781e7	−1.48451
$977$	5.01796e7i	1.68186i	0.541142	+	0.840931i	$0.317992\pi$
$977$	5.01796e7i	1.68186i	−0.541142	+	0.840931i	$0.682008\pi$
$978$	− 660352.i	− 0.0220764i
$979$	2.26020e7	0.753684
$980$	0	0
$981$	368979.	0.0122414
$982$	1.85349e6i	0.0613355i
$983$	1.44682e7i	0.477563i	0.971073	+	0.238781i	$0.0767479\pi$
$983$	1.44682e7i	0.477563i	−0.971073	+	0.238781i	$0.923252\pi$
$984$	2.93358e6	0.0965852
$985$	0	0
$986$	−101481.	−0.00332425
$987$	1.20267e6i	0.0392964i
$988$	6.34394e6i	0.206760i
$989$	−1.81743e7	−0.590836
$990$	0	0
$991$	2.92271e7	0.945369	0.472685	−	0.881232i	$-0.343285\pi$
$991$	2.92271e7	0.945369	0.472685	+	0.881232i	$0.343285\pi$
$992$	− 9.51250e6i	− 0.306913i
$993$	2.59080e7i	0.833799i
$994$	41586.6	0.00133502
$995$	0	0
$996$	−3.15573e7	−1.00798
$997$	3.35666e7i	1.06947i	0.845019	+	0.534736i	$0.179589\pi$
$997$	3.35666e7i	1.06947i	−0.845019	+	0.534736i	$0.820411\pi$
$998$	4.04595e6i	0.128586i
$999$	6.97972e6	0.221271

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.h.274.9		18
5.2	odd	4		325.6.a.i.1.5	yes	9
5.3	odd	4		325.6.a.h.1.5	✓	9
5.4	even	2	inner	325.6.b.h.274.10		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.h.1.5	✓	9	5.3	odd	4
325.6.a.i.1.5	yes	9	5.2	odd	4
325.6.b.h.274.9		18	1.1	even	1	trivial
325.6.b.h.274.10		18	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.396608i q^{2} -11.4974i q^{3} +31.8427 q^{4} -4.55994 q^{6} -8.04695i q^{7} -25.3205i q^{8} +110.811 q^{9} +O(q^{10})\)	\(q-0.396608i q^{2} -11.4974i q^{3} +31.8427 q^{4} -4.55994 q^{6} -8.04695i q^{7} -25.3205i q^{8} +110.811 q^{9} +335.903 q^{11} -366.107i q^{12} -169.000i q^{13} -3.19148 q^{14} +1008.92 q^{16} -38.1592i q^{17} -43.9484i q^{18} -1178.86 q^{19} -92.5186 q^{21} -133.222i q^{22} +1679.35i q^{23} -291.119 q^{24} -67.0267 q^{26} -4067.89i q^{27} -256.237i q^{28} +6705.42 q^{29} +7858.95 q^{31} -1210.40i q^{32} -3861.99i q^{33} -15.1342 q^{34} +3528.51 q^{36} +1715.81i q^{37} +467.545i q^{38} -1943.05 q^{39} -10076.9 q^{41} +36.6936i q^{42} +10822.3i q^{43} +10696.0 q^{44} +666.042 q^{46} -12999.2i q^{47} -11600.0i q^{48} +16742.2 q^{49} -438.730 q^{51} -5381.42i q^{52} +2639.19i q^{53} -1613.36 q^{54} -203.753 q^{56} +13553.8i q^{57} -2659.42i q^{58} -4956.81 q^{59} -43787.4 q^{61} -3116.92i q^{62} -891.687i q^{63} +31805.5 q^{64} -1531.70 q^{66} +9170.90i q^{67} -1215.09i q^{68} +19308.0 q^{69} -13030.5 q^{71} -2805.78i q^{72} -41726.8i q^{73} +680.503 q^{74} -37538.1 q^{76} -2702.99i q^{77} +770.630i q^{78} +49809.7 q^{79} -19843.0 q^{81} +3996.58i q^{82} -86196.9i q^{83} -2946.04 q^{84} +4292.19 q^{86} -77094.6i q^{87} -8505.22i q^{88} +67287.2 q^{89} -1359.93 q^{91} +53474.9i q^{92} -90357.2i q^{93} -5155.58 q^{94} -13916.4 q^{96} -175537. i q^{97} -6640.10i q^{98} +37221.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9}+O(q^{10}) \) 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9	\( 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9} - 2844 q^{11} + 684 q^{14} - 2122 q^{16} + 816 q^{19} - 7824 q^{21} + 16938 q^{24} - 1690 q^{26} + 17474 q^{29} + 1496 q^{31} + 35578 q^{34} + 1024 q^{36} + 3718 q^{39} - 57352 q^{41} + 61198 q^{44} - 24112 q^{46} + 81814 q^{49} - 62012 q^{51} + 205522 q^{54} - 46004 q^{56} + 176284 q^{59} + 56330 q^{61} + 201690 q^{64} + 85154 q^{66} + 363494 q^{69} - 141124 q^{71} + 271352 q^{74} + 92746 q^{76} + 328146 q^{79} - 139870 q^{81} + 691312 q^{84} - 589840 q^{86} + 505396 q^{89} + 4056 q^{91} + 1003212 q^{94} - 638362 q^{96} + 1515552 q^{99}+O(q^{100}) \) 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9 - 2844 * q^11 + 684 * q^14 - 2122 * q^16 + 816 * q^19 - 7824 * q^21 + 16938 * q^24 - 1690 * q^26 + 17474 * q^29 + 1496 * q^31 + 35578 * q^34 + 1024 * q^36 + 3718 * q^39 - 57352 * q^41 + 61198 * q^44 - 24112 * q^46 + 81814 * q^49 - 62012 * q^51 + 205522 * q^54 - 46004 * q^56 + 176284 * q^59 + 56330 * q^61 + 201690 * q^64 + 85154 * q^66 + 363494 * q^69 - 141124 * q^71 + 271352 * q^74 + 92746 * q^76 + 328146 * q^79 - 139870 * q^81 + 691312 * q^84 - 589840 * q^86 + 505396 * q^89 + 4056 * q^91 + 1003212 * q^94 - 638362 * q^96 + 1515552 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more