LMFDB - Embedded newform 325.6.b.g.274.2

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:	$12$
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} + 326x^{10} + 38809x^{8} + 2034064x^{6} + 43897824x^{4} + 281822976x^{2} + 377913600 $ x^12 + 326x^10 + 38809x^8 + 2034064x^6 + 43897824x^4 + 281822976*x^2 + 377913600
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.2
Root			$-9.61672i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.6.b.g.274.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-9.61672i q^{2} +28.9791i q^{3} -60.4813 q^{4} +278.684 q^{6} +189.995i q^{7} +273.896i q^{8} -596.790 q^{9} -566.944 q^{11} -1752.69i q^{12} +169.000i q^{13} +1827.13 q^{14} +698.582 q^{16} +686.585i q^{17} +5739.16i q^{18} -125.865 q^{19} -5505.88 q^{21} +5452.14i q^{22} +3481.79i q^{23} -7937.27 q^{24} +1625.23 q^{26} -10252.5i q^{27} -11491.1i q^{28} -1909.26 q^{29} +6579.43 q^{31} +2046.61i q^{32} -16429.5i q^{33} +6602.70 q^{34} +36094.6 q^{36} -6923.49i q^{37} +1210.41i q^{38} -4897.47 q^{39} -1117.84 q^{41} +52948.5i q^{42} -3465.64i q^{43} +34289.5 q^{44} +33483.4 q^{46} -7747.01i q^{47} +20244.3i q^{48} -19291.0 q^{49} -19896.6 q^{51} -10221.3i q^{52} -15186.2i q^{53} -98595.7 q^{54} -52038.8 q^{56} -3647.45i q^{57} +18360.8i q^{58} +48418.6 q^{59} -14031.1 q^{61} -63272.5i q^{62} -113387. i q^{63} +42036.3 q^{64} -157998. q^{66} +33089.4i q^{67} -41525.5i q^{68} -100899. q^{69} +21014.6 q^{71} -163459. i q^{72} -83446.8i q^{73} -66581.3 q^{74} +7612.47 q^{76} -107716. i q^{77} +47097.6i q^{78} -74858.5 q^{79} +152089. q^{81} +10750.0i q^{82} +51860.8i q^{83} +333003. q^{84} -33328.1 q^{86} -55328.7i q^{87} -155284. i q^{88} -122582. q^{89} -32109.1 q^{91} -210583. i q^{92} +190666. i q^{93} -74500.8 q^{94} -59309.0 q^{96} -19305.3i q^{97} +185516. i q^{98} +338346. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 268 q^{4} + 636 q^{6} - 1036 q^{9} - 340 q^{11} + 2880 q^{14} + 7012 q^{16} - 2436 q^{19} - 792 q^{21} - 25236 q^{24} - 16728 q^{29} + 5724 q^{31} + 42968 q^{34} - 4276 q^{36} - 12844 q^{39} + 4496 q^{41}+ \cdots + 64540 q^{99}+O(q^{100}) $ 12 * q - 268 * q^4 + 636 * q^6 - 1036 * q^9 - 340 * q^11 + 2880 * q^14 + 7012 * q^16 - 2436 * q^19 - 792 * q^21 - 25236 * q^24 - 16728 * q^29 + 5724 * q^31 + 42968 * q^34 - 4276 * q^36 - 12844 * q^39 + 4496 * q^41 + 146964 * q^44 - 79772 * q^46 - 171804 * q^49 - 110056 * q^51 - 91008 * q^54 - 377600 * q^56 + 57748 * q^59 + 1944 * q^61 - 28044 * q^64 - 316080 * q^66 - 306176 * q^69 - 148348 * q^71 - 452224 * q^74 - 186844 * q^76 - 429016 * q^79 + 232012 * q^81 + 470536 * q^84 - 609332 * q^86 - 188056 * q^89 + 74360 * q^91 + 251840 * q^94 - 771716 * q^96 + 64540 * q^99

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 $

$2$	− 9.61672i	− 1.70001i	−0.526773	−	0.850006i	$-0.676598\pi$
$2$	− 9.61672i	− 1.70001i	0.526773	−	0.850006i	$-0.323402\pi$
$3$	28.9791i	1.85901i	0.368807	+	0.929506i	$0.379766\pi$
$3$	28.9791i	1.85901i	−0.368807	+	0.929506i	$0.620234\pi$
$4$	−60.4813	−1.89004
$5$	0	0
$5$	0	0
$6$	278.684	3.16034
$7$	189.995i	1.46554i	0.680478	+	0.732768i	$0.261772\pi$
$7$	189.995i	1.46554i	−0.680478	+	0.732768i	$0.738228\pi$
$8$	273.896i	1.51308i
$9$	−596.790	−2.45593
$10$	0	0
$11$	−566.944	−1.41273	−0.706363	−	0.707849i	$-0.749666\pi$
$11$	−566.944	−1.41273	−0.706363	+	0.707849i	$0.749666\pi$
$12$	− 1752.69i	− 3.51361i
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	1827.13	2.49143
$15$	0	0
$16$	698.582	0.682209
$17$	686.585i	0.576198i	0.957601	+	0.288099i	$0.0930233\pi$
$17$	686.585i	0.576198i	−0.957601	+	0.288099i	$0.906977\pi$
$18$	5739.16i	4.17510i
$19$	−125.865	−0.0799872	−0.0399936	−	0.999200i	$-0.512734\pi$
$19$	−125.865	−0.0799872	−0.0399936	+	0.999200i	$0.512734\pi$
$20$	0	0
$21$	−5505.88	−2.72445
$22$	5452.14i	2.40165i
$23$	3481.79i	1.37241i	0.727410	+	0.686203i	$0.240724\pi$
$23$	3481.79i	1.37241i	−0.727410	+	0.686203i	$0.759276\pi$
$24$	−7937.27	−2.81283
$25$	0	0
$26$	1625.23	0.471498
$27$	− 10252.5i	− 2.70659i
$28$	− 11491.1i	− 2.76992i
$29$	−1909.26	−0.421570	−0.210785	−	0.977532i	$-0.567602\pi$
$29$	−1909.26	−0.421570	−0.210785	+	0.977532i	$0.567602\pi$
$30$	0	0
$31$	6579.43	1.22966	0.614828	−	0.788661i	$-0.289225\pi$
$31$	6579.43	1.22966	0.614828	+	0.788661i	$0.289225\pi$
$32$	2046.61i	0.353314i
$33$	− 16429.5i	− 2.62628i
$34$	6602.70	0.979544
$35$	0	0
$36$	36094.6	4.64180
$37$	− 6923.49i	− 0.831421i	−0.909497	−	0.415710i	$-0.863533\pi$
$37$	− 6923.49i	− 0.831421i	0.909497	−	0.415710i	$-0.136467\pi$
$38$	1210.41i	0.135979i
$39$	−4897.47	−0.515597
$40$	0	0
$41$	−1117.84	−0.103853	−0.0519266	−	0.998651i	$-0.516536\pi$
$41$	−1117.84	−0.103853	−0.0519266	+	0.998651i	$0.516536\pi$
$42$	52948.5i	4.63160i
$43$	− 3465.64i	− 0.285833i	−0.989735	−	0.142916i	$-0.954352\pi$
$43$	− 3465.64i	− 0.285833i	0.989735	−	0.142916i	$-0.0456480\pi$
$44$	34289.5	2.67011
$45$	0	0
$46$	33483.4	2.33311
$47$	− 7747.01i	− 0.511552i	−0.966736	−	0.255776i	$-0.917669\pi$
$47$	− 7747.01i	− 0.511552i	0.966736	−	0.255776i	$-0.0823309\pi$
$48$	20244.3i	1.26824i
$49$	−19291.0	−1.14780
$50$	0	0
$51$	−19896.6	−1.07116
$52$	− 10221.3i	− 0.524203i
$53$	− 15186.2i	− 0.742609i	−0.928511	−	0.371305i	$-0.878911\pi$
$53$	− 15186.2i	− 0.742609i	0.928511	−	0.371305i	$-0.121089\pi$
$54$	−98595.7	−4.60123
$55$	0	0
$56$	−52038.8	−2.21747
$57$	− 3647.45i	− 0.148697i
$58$	18360.8i	0.716675i
$59$	48418.6	1.81085	0.905424	−	0.424508i	$-0.139553\pi$
$59$	48418.6	1.81085	0.905424	+	0.424508i	$0.139553\pi$
$60$	0	0
$61$	−14031.1	−0.482801	−0.241400	−	0.970426i	$-0.577607\pi$
$61$	−14031.1	−0.482801	−0.241400	+	0.970426i	$0.577607\pi$
$62$	− 63272.5i	− 2.09043i
$63$	− 113387.i	− 3.59925i
$64$	42036.3	1.28285
$65$	0	0
$66$	−157998.	−4.46470
$67$	33089.4i	0.900537i	0.892893	+	0.450268i	$0.148672\pi$
$67$	33089.4i	0.900537i	−0.892893	+	0.450268i	$0.851328\pi$
$68$	− 41525.5i	− 1.08904i
$69$	−100899.	−2.55132
$70$	0	0
$71$	21014.6	0.494738	0.247369	−	0.968921i	$-0.420434\pi$
$71$	21014.6	0.494738	0.247369	+	0.968921i	$0.420434\pi$
$72$	− 163459.i	− 3.71601i
$73$	− 83446.8i	− 1.83275i	−0.400325	−	0.916373i	$-0.631103\pi$
$73$	− 83446.8i	− 1.83275i	0.400325	−	0.916373i	$-0.368897\pi$
$74$	−66581.3	−1.41342
$75$	0	0
$76$	7612.47	0.151179
$77$	− 107716.i	− 2.07040i
$78$	47097.6i	0.876521i
$79$	−74858.5	−1.34950	−0.674751	−	0.738045i	$-0.735749\pi$
$79$	−74858.5	−1.34950	−0.674751	+	0.738045i	$0.735749\pi$
$80$	0	0
$81$	152089.	2.57565
$82$	10750.0i	0.176552i
$83$	51860.8i	0.826311i	0.910660	+	0.413156i	$0.135573\pi$
$83$	51860.8i	0.826311i	−0.910660	+	0.413156i	$0.864427\pi$
$84$	333003.	5.14932
$85$	0	0
$86$	−33328.1	−0.485919
$87$	− 55328.7i	− 0.783705i
$88$	− 155284.i	− 2.13756i
$89$	−122582.	−1.64041	−0.820205	−	0.572070i	$-0.806141\pi$
$89$	−122582.	−1.64041	−0.820205	+	0.572070i	$0.806141\pi$
$90$	0	0
$91$	−32109.1	−0.406467
$92$	− 210583.i	− 2.59390i
$93$	190666.i	2.28595i
$94$	−74500.8	−0.869644
$95$	0	0
$96$	−59309.0	−0.656814
$97$	− 19305.3i	− 0.208328i	−0.994560	−	0.104164i	$-0.966783\pi$
$97$	− 19305.3i	− 0.208328i	0.994560	−	0.104164i	$-0.0332166\pi$
$98$	185516.i	1.95127i
$99$	338346.	3.46955
$100$	0	0
$101$	−83222.2	−0.811775	−0.405888	−	0.913923i	$-0.633038\pi$
$101$	−83222.2	−0.811775	−0.405888	+	0.913923i	$0.633038\pi$
$102$	191340.i	1.82098i
$103$	109529.i	1.01727i	0.860983	+	0.508634i	$0.169849\pi$
$103$	109529.i	1.01727i	−0.860983	+	0.508634i	$0.830151\pi$
$104$	−46288.5	−0.419652
$105$	0	0
$106$	−146042.	−1.26244
$107$	65171.1i	0.550295i	0.961402	+	0.275148i	$0.0887267\pi$
$107$	65171.1i	0.550295i	−0.961402	+	0.275148i	$0.911273\pi$
$108$	620086.i	5.11555i
$109$	180012.	1.45123	0.725615	−	0.688101i	$-0.241555\pi$
$109$	180012.	1.45123	0.725615	+	0.688101i	$0.241555\pi$
$110$	0	0
$111$	200637.	1.54562
$112$	132727.i	0.999802i
$113$	− 63753.0i	− 0.469683i	−0.972034	−	0.234841i	$-0.924543\pi$
$113$	− 63753.0i	− 0.469683i	0.972034	−	0.234841i	$-0.0754571\pi$
$114$	−35076.5	−0.252787
$115$	0	0
$116$	115474.	0.796785
$117$	− 100858.i	− 0.681151i
$118$	− 465628.i	− 3.07846i
$119$	−130448.	−0.844440
$120$	0	0
$121$	160374.	0.995797
$122$	134933.i	0.820767i
$123$	− 32394.0i	− 0.193064i
$124$	−397932.	−2.32410
$125$	0	0
$126$	−1.09041e6	−6.11877
$127$	18792.2i	0.103388i	0.998663	+	0.0516939i	$0.0164620\pi$
$127$	18792.2i	0.103388i	−0.998663	+	0.0516939i	$0.983538\pi$
$128$	− 338760.i	− 1.82754i
$129$	100431.	0.531367
$130$	0	0
$131$	−18190.6	−0.0926122	−0.0463061	−	0.998927i	$-0.514745\pi$
$131$	−18190.6	−0.0926122	−0.0463061	+	0.998927i	$0.514745\pi$
$132$	993679.i	4.96377i
$133$	− 23913.7i	− 0.117224i
$134$	318211.	1.53092
$135$	0	0
$136$	−188053.	−0.871833
$137$	− 106188.i	− 0.483363i	−0.970356	−	0.241682i	$-0.922301\pi$
$137$	− 106188.i	− 0.483363i	0.970356	−	0.241682i	$-0.0776990\pi$
$138$	970319.i	4.33727i
$139$	−33944.0	−0.149014	−0.0745069	−	0.997221i	$-0.523738\pi$
$139$	−33944.0	−0.149014	−0.0745069	+	0.997221i	$0.523738\pi$
$140$	0	0
$141$	224502.	0.950981
$142$	− 202092.i	− 0.841061i
$143$	− 95813.5i	− 0.391820i
$144$	−416907.	−1.67546
$145$	0	0
$146$	−802484.	−3.11569
$147$	− 559037.i	− 2.13377i
$148$	418741.i	1.57142i
$149$	−248671.	−0.917612	−0.458806	−	0.888536i	$-0.651723\pi$
$149$	−248671.	−0.917612	−0.458806	+	0.888536i	$0.651723\pi$
$150$	0	0
$151$	62344.4	0.222513	0.111256	−	0.993792i	$-0.464512\pi$
$151$	62344.4	0.222513	0.111256	+	0.993792i	$0.464512\pi$
$152$	− 34473.9i	− 0.121027i
$153$	− 409747.i	− 1.41510i
$154$	−1.03588e6	−3.51971
$155$	0	0
$156$	296205.	0.974499
$157$	275219.i	0.891106i	0.895256	+	0.445553i	$0.146993\pi$
$157$	275219.i	0.891106i	−0.895256	+	0.445553i	$0.853007\pi$
$158$	719893.i	2.29417i
$159$	440084.	1.38052
$160$	0	0
$161$	−661521.	−2.01131
$162$	− 1.46260e6i	− 4.37863i
$163$	− 53135.9i	− 0.156646i	−0.996928	−	0.0783229i	$-0.975043\pi$
$163$	− 53135.9i	− 0.156646i	0.996928	−	0.0783229i	$-0.0249565\pi$
$164$	67608.4	0.196287
$165$	0	0
$166$	498730.	1.40474
$167$	− 412563.i	− 1.14472i	−0.820002	−	0.572360i	$-0.806028\pi$
$167$	− 412563.i	− 1.14472i	0.820002	−	0.572360i	$-0.193972\pi$
$168$	− 1.50804e6i	− 4.12230i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	75114.9	0.196443
$172$	209606.i	0.540235i
$173$	− 584248.i	− 1.48416i	−0.670309	−	0.742082i	$-0.733838\pi$
$173$	− 584248.i	− 1.48416i	0.670309	−	0.742082i	$-0.266162\pi$
$174$	−532081.	−1.33231
$175$	0	0
$176$	−396057.	−0.963775
$177$	1.40313e6i	3.36639i
$178$	1.17884e6i	2.78872i
$179$	19918.7	0.0464652	0.0232326	−	0.999730i	$-0.492604\pi$
$179$	19918.7	0.0464652	0.0232326	+	0.999730i	$0.492604\pi$
$180$	0	0
$181$	585612.	1.32866	0.664329	−	0.747440i	$-0.268717\pi$
$181$	585612.	1.32866	0.664329	+	0.747440i	$0.268717\pi$
$182$	308784.i	0.690998i
$183$	− 406610.i	− 0.897533i
$184$	−953648.	−2.07656
$185$	0	0
$186$	1.83358e6	3.88614
$187$	− 389255.i	− 0.814011i
$188$	468549.i	0.966853i
$189$	1.94793e6	3.96660
$190$	0	0
$191$	−257249.	−0.510235	−0.255117	−	0.966910i	$-0.582114\pi$
$191$	−257249.	−0.510235	−0.255117	+	0.966910i	$0.582114\pi$
$192$	1.21818e6i	2.38483i
$193$	729315.i	1.40936i	0.709525	+	0.704680i	$0.248909\pi$
$193$	729315.i	1.40936i	−0.709525	+	0.704680i	$0.751091\pi$
$194$	−185653.	−0.354159
$195$	0	0
$196$	1.16674e6	2.16938
$197$	− 614173.i	− 1.12752i	−0.825938	−	0.563761i	$-0.809354\pi$
$197$	− 614173.i	− 1.12752i	0.825938	−	0.563761i	$-0.190646\pi$
$198$	− 3.25378e6i	− 5.89828i
$199$	2188.99	0.00391842	0.00195921	−	0.999998i	$-0.499376\pi$
$199$	2188.99	0.00391842	0.00195921	+	0.999998i	$0.499376\pi$
$200$	0	0
$201$	−958901.	−1.67411
$202$	800325.i	1.38003i
$203$	− 362749.i	− 0.617827i
$204$	1.20337e6	2.02453
$205$	0	0
$206$	1.05331e6	1.72937
$207$	− 2.07790e6i	− 3.37053i
$208$	118060.i	0.189211i
$209$	71358.3	0.113000
$210$	0	0
$211$	95873.8	0.148250	0.0741248	−	0.997249i	$-0.476384\pi$
$211$	95873.8	0.148250	0.0741248	+	0.997249i	$0.476384\pi$
$212$	918483.i	1.40356i
$213$	608985.i	0.919725i
$214$	626732.	0.935508
$215$	0	0
$216$	2.80813e6	4.09527
$217$	1.25006e6i	1.80211i
$218$	− 1.73113e6i	− 2.46711i
$219$	2.41822e6	3.40710
$220$	0	0
$221$	−116033.	−0.159809
$222$	− 1.92947e6i	− 2.62757i
$223$	822496.i	1.10757i	0.832659	+	0.553786i	$0.186817\pi$
$223$	822496.i	1.10757i	−0.832659	+	0.553786i	$0.813183\pi$
$224$	−388845.	−0.517794
$225$	0	0
$226$	−613095.	−0.798466
$227$	431133.i	0.555325i	0.960679	+	0.277662i	$0.0895597\pi$
$227$	431133.i	0.555325i	−0.960679	+	0.277662i	$0.910440\pi$
$228$	220603.i	0.281044i
$229$	−962073.	−1.21233	−0.606163	−	0.795341i	$-0.707292\pi$
$229$	−962073.	−1.21233	−0.606163	+	0.795341i	$0.707292\pi$
$230$	0	0
$231$	3.12153e6	3.84890
$232$	− 522939.i	− 0.637869i
$233$	− 517023.i	− 0.623907i	−0.950097	−	0.311953i	$-0.899017\pi$
$233$	− 517023.i	− 0.623907i	0.950097	−	0.311953i	$-0.100983\pi$
$234$	−969918.	−1.15797
$235$	0	0
$236$	−2.92842e6	−3.42257
$237$	− 2.16934e6i	− 2.50874i
$238$	1.25448e6i	1.43556i
$239$	573574.	0.649523	0.324761	−	0.945796i	$-0.394716\pi$
$239$	573574.	0.649523	0.324761	+	0.945796i	$0.394716\pi$
$240$	0	0
$241$	−967200.	−1.07269	−0.536344	−	0.843999i	$-0.680195\pi$
$241$	−967200.	−1.07269	−0.536344	+	0.843999i	$0.680195\pi$
$242$	− 1.54227e6i	− 1.69287i
$243$	1.91606e6i	2.08158i
$244$	848620.	0.912513
$245$	0	0
$246$	−311524.	−0.328212
$247$	− 21271.2i	− 0.0221845i
$248$	1.80208e6i	1.86057i
$249$	−1.50288e6	−1.53612
$250$	0	0
$251$	−1.51519e6	−1.51804	−0.759018	−	0.651069i	$-0.774321\pi$
$251$	−1.51519e6	−1.51804	−0.759018	+	0.651069i	$0.774321\pi$
$252$	6.85779e6i	6.80272i
$253$	− 1.97398e6i	− 1.93883i
$254$	180720.	0.175760
$255$	0	0
$256$	−1.91259e6	−1.82399
$257$	− 408239.i	− 0.385551i	−0.981243	−	0.192775i	$-0.938251\pi$
$257$	− 408239.i	− 0.385551i	0.981243	−	0.192775i	$-0.0617489\pi$
$258$	− 965819.i	− 0.903330i
$259$	1.31543e6	1.21848
$260$	0	0
$261$	1.13943e6	1.03535
$262$	174934.i	0.157442i
$263$	− 1.24387e6i	− 1.10888i	−0.832224	−	0.554440i	$-0.812932\pi$
$263$	− 1.24387e6i	− 1.10888i	0.832224	−	0.554440i	$-0.187068\pi$
$264$	4.49999e6	3.97376
$265$	0	0
$266$	−229971.	−0.199282
$267$	− 3.55233e6i	− 3.04954i
$268$	− 2.00129e6i	− 1.70205i
$269$	971010.	0.818169	0.409084	−	0.912497i	$-0.365848\pi$
$269$	971010.	0.818169	0.409084	+	0.912497i	$0.365848\pi$
$270$	0	0
$271$	647163.	0.535292	0.267646	−	0.963517i	$-0.413754\pi$
$271$	647163.	0.535292	0.267646	+	0.963517i	$0.413754\pi$
$272$	479636.i	0.393088i
$273$	− 930494.i	− 0.755626i
$274$	−1.02118e6	−0.821724
$275$	0	0
$276$	6.10251e6	4.82209
$277$	− 684624.i	− 0.536109i	−0.963404	−	0.268054i	$-0.913619\pi$
$277$	− 684624.i	− 0.536109i	0.963404	−	0.268054i	$-0.0863807\pi$
$278$	326430.i	0.253325i
$279$	−3.92654e6	−3.01995
$280$	0	0
$281$	−2.08097e6	−1.57217	−0.786085	−	0.618118i	$-0.787895\pi$
$281$	−2.08097e6	−1.57217	−0.786085	+	0.618118i	$0.787895\pi$
$282$	− 2.15897e6i	− 1.61668i
$283$	2.09568e6i	1.55546i	0.628598	+	0.777730i	$0.283629\pi$
$283$	2.09568e6i	1.55546i	−0.628598	+	0.777730i	$0.716371\pi$
$284$	−1.27099e6	−0.935075
$285$	0	0
$286$	−921411.	−0.666098
$287$	− 212384.i	− 0.152201i
$288$	− 1.22140e6i	− 0.867712i
$289$	948458.	0.667995
$290$	0	0
$291$	559450.	0.387284
$292$	5.04697e6i	3.46396i
$293$	170648.i	0.116127i	0.998313	+	0.0580633i	$0.0184925\pi$
$293$	170648.i	0.116127i	−0.998313	+	0.0580633i	$0.981507\pi$
$294$	−5.37610e6	−3.62743
$295$	0	0
$296$	1.89632e6	1.25800
$297$	5.81261e6i	3.82367i
$298$	2.39140e6i	1.55995i
$299$	−588422.	−0.380637
$300$	0	0
$301$	658453.	0.418898
$302$	− 599549.i	− 0.378275i
$303$	− 2.41171e6i	− 1.50910i
$304$	−87926.9	−0.0545680
$305$	0	0
$306$	−3.94042e6	−2.40569
$307$	42275.0i	0.0255998i	0.999918	+	0.0127999i	$0.00407445\pi$
$307$	42275.0i	0.0255998i	−0.999918	+	0.0127999i	$0.995926\pi$
$308$	6.51482e6i	3.91314i
$309$	−3.17405e6	−1.89111
$310$	0	0
$311$	−1.35058e6	−0.791808	−0.395904	−	0.918292i	$-0.629569\pi$
$311$	−1.35058e6	−0.791808	−0.395904	+	0.918292i	$0.629569\pi$
$312$	− 1.34140e6i	− 0.780138i
$313$	2.22900e6i	1.28603i	0.765856	+	0.643013i	$0.222316\pi$
$313$	2.22900e6i	1.28603i	−0.765856	+	0.643013i	$0.777684\pi$
$314$	2.64670e6	1.51489
$315$	0	0
$316$	4.52754e6	2.55061
$317$	3.24898e6i	1.81593i	0.419048	+	0.907964i	$0.362364\pi$
$317$	3.24898e6i	1.81593i	−0.419048	+	0.907964i	$0.637636\pi$
$318$	− 4.23216e6i	− 2.34690i
$319$	1.08244e6	0.595564
$320$	0	0
$321$	−1.88860e6	−1.02301
$322$	6.36166e6i	3.41925i
$323$	− 86416.9i	− 0.0460885i
$324$	−9.19856e6	−4.86808
$325$	0	0
$326$	−510993.	−0.266300
$327$	5.21661e6i	2.69786i
$328$	− 306172.i	− 0.157138i
$329$	1.47189e6	0.749698
$330$	0	0
$331$	1.81129e6	0.908694	0.454347	−	0.890825i	$-0.349873\pi$
$331$	1.81129e6	0.908694	0.454347	+	0.890825i	$0.349873\pi$
$332$	− 3.13660e6i	− 1.56176i
$333$	4.13187e6i	2.04191i
$334$	−3.96750e6	−1.94604
$335$	0	0
$336$	−3.84631e6	−1.85864
$337$	1.43117e6i	0.686461i	0.939251	+	0.343231i	$0.111521\pi$
$337$	1.43117e6i	0.686461i	−0.939251	+	0.343231i	$0.888479\pi$
$338$	274663.i	0.130770i
$339$	1.84751e6	0.873146
$340$	0	0
$341$	−3.73016e6	−1.73717
$342$	− 722359.i	− 0.333955i
$343$	− 471949.i	− 0.216601i
$344$	949225.	0.432487
$345$	0	0
$346$	−5.61855e6	−2.52310
$347$	− 2.89201e6i	− 1.28937i	−0.764450	−	0.644683i	$-0.776989\pi$
$347$	− 2.89201e6i	− 1.28937i	0.764450	−	0.644683i	$-0.223011\pi$
$348$	3.34635e6i	1.48123i
$349$	3.29833e6	1.44954	0.724771	−	0.688990i	$-0.241946\pi$
$349$	3.29833e6	1.44954	0.724771	+	0.688990i	$0.241946\pi$
$350$	0	0
$351$	1.73268e6	0.750672
$352$	− 1.16031e6i	− 0.499136i
$353$	2.87329e6i	1.22728i	0.789587	+	0.613639i	$0.210295\pi$
$353$	2.87329e6i	1.22728i	−0.789587	+	0.613639i	$0.789705\pi$
$354$	1.34935e7	5.72290
$355$	0	0
$356$	7.41393e6	3.10044
$357$	− 3.78026e6i	− 1.56982i
$358$	− 191552.i	− 0.0789914i
$359$	655112.	0.268274	0.134137	−	0.990963i	$-0.457174\pi$
$359$	655112.	0.268274	0.134137	+	0.990963i	$0.457174\pi$
$360$	0	0
$361$	−2.46026e6	−0.993602
$362$	− 5.63166e6i	− 2.25873i
$363$	4.64750e6i	1.85120i
$364$	1.94200e6	0.768238
$365$	0	0
$366$	−3.91025e6	−1.52582
$367$	− 3.46555e6i	− 1.34310i	−0.740960	−	0.671549i	$-0.765629\pi$
$367$	− 3.46555e6i	− 1.34310i	0.740960	−	0.671549i	$-0.234371\pi$
$368$	2.43231e6i	0.936268i
$369$	667116.	0.255056
$370$	0	0
$371$	2.88531e6	1.08832
$372$	− 1.15317e7i	− 4.32053i
$373$	3.97540e6i	1.47948i	0.672894	+	0.739739i	$0.265051\pi$
$373$	3.97540e6i	1.47948i	−0.672894	+	0.739739i	$0.734949\pi$
$374$	−3.74336e6	−1.38383
$375$	0	0
$376$	2.12188e6	0.774017
$377$	− 322665.i	− 0.116923i
$378$	− 1.87327e7i	− 6.74326i
$379$	187178.	0.0669357	0.0334679	−	0.999440i	$-0.489345\pi$
$379$	187178.	0.0669357	0.0334679	+	0.999440i	$0.489345\pi$
$380$	0	0
$381$	−544583.	−0.192199
$382$	2.47389e6i	0.867405i
$383$	− 2.85416e6i	− 0.994216i	−0.867689	−	0.497108i	$-0.834395\pi$
$383$	− 2.85416e6i	− 0.994216i	0.867689	−	0.497108i	$-0.165605\pi$
$384$	9.81696e6	3.39742
$385$	0	0
$386$	7.01362e6	2.39593
$387$	2.06826e6i	0.701985i
$388$	1.16761e6i	0.393747i
$389$	3.54177e6	1.18671	0.593357	−	0.804939i	$-0.297802\pi$
$389$	3.54177e6	1.18671	0.593357	+	0.804939i	$0.297802\pi$
$390$	0	0
$391$	−2.39054e6	−0.790778
$392$	− 5.28373e6i	− 1.73670i
$393$	− 527147.i	− 0.172167i
$394$	−5.90633e6	−1.91680
$395$	0	0
$396$	−2.04636e7	−6.55759
$397$	1.24829e6i	0.397501i	0.980050	+	0.198751i	$0.0636884\pi$
$397$	1.24829e6i	0.397501i	−0.980050	+	0.198751i	$0.936312\pi$
$398$	− 21050.9i	− 0.00666136i
$399$	692997.	0.217921
$400$	0	0
$401$	−2.74122e6	−0.851299	−0.425650	−	0.904888i	$-0.639954\pi$
$401$	−2.74122e6	−0.851299	−0.425650	+	0.904888i	$0.639954\pi$
$402$	9.22148e6i	2.84600i
$403$	1.11192e6i	0.341045i
$404$	5.03339e6	1.53429
$405$	0	0
$406$	−3.48846e6	−1.05031
$407$	3.92523e6i	1.17457i
$408$	− 5.44961e6i	− 1.62075i
$409$	−6.13745e6	−1.81418	−0.907089	−	0.420940i	$-0.861700\pi$
$409$	−6.13745e6	−1.81418	−0.907089	+	0.420940i	$0.861700\pi$
$410$	0	0
$411$	3.07723e6	0.898579
$412$	− 6.62444e6i	− 1.92267i
$413$	9.19928e6i	2.65386i
$414$	−1.99825e7	−5.72994
$415$	0	0
$416$	−345877.	−0.0979916
$417$	− 983668.i	− 0.277018i
$418$	− 686232.i	− 0.192101i
$419$	−1.95897e6	−0.545120	−0.272560	−	0.962139i	$-0.587870\pi$
$419$	−1.95897e6	−0.545120	−0.272560	+	0.962139i	$0.587870\pi$
$420$	0	0
$421$	5.26629e6	1.44810	0.724051	−	0.689746i	$-0.242278\pi$
$421$	5.26629e6	1.44810	0.724051	+	0.689746i	$0.242278\pi$
$422$	− 921991.i	− 0.252026i
$423$	4.62334e6i	1.25633i
$424$	4.15945e6	1.12363
$425$	0	0
$426$	5.85644e6	1.56354
$427$	− 2.66584e6i	− 0.707562i
$428$	− 3.94163e6i	− 1.04008i
$429$	2.77659e6	0.728398
$430$	0	0
$431$	−3.23336e6	−0.838420	−0.419210	−	0.907889i	$-0.637693\pi$
$431$	−3.23336e6	−0.838420	−0.419210	+	0.907889i	$0.637693\pi$
$432$	− 7.16224e6i	− 1.84646i
$433$	− 3.49771e6i	− 0.896529i	−0.893901	−	0.448265i	$-0.852042\pi$
$433$	− 3.49771e6i	− 0.896529i	0.893901	−	0.448265i	$-0.147958\pi$
$434$	1.20214e7	3.06360
$435$	0	0
$436$	−1.08874e7	−2.74288
$437$	− 438235.i	− 0.109775i
$438$	− 2.32553e7i	− 5.79211i
$439$	29404.2	0.00728197	0.00364098	−	0.999993i	$-0.498841\pi$
$439$	29404.2	0.00728197	0.00364098	+	0.999993i	$0.498841\pi$
$440$	0	0
$441$	1.15127e7	2.81890
$442$	1.11586e6i	0.271677i
$443$	545860.i	0.132151i	0.997815	+	0.0660757i	$0.0210479\pi$
$443$	545860.i	0.132151i	−0.997815	+	0.0660757i	$0.978952\pi$
$444$	−1.21348e7	−2.92128
$445$	0	0
$446$	7.90971e6	1.88288
$447$	− 7.20626e6i	− 1.70585i
$448$	7.98668e6i	1.88006i
$449$	5.46505e6	1.27932	0.639658	−	0.768660i	$-0.279076\pi$
$449$	5.46505e6	1.27932	0.639658	+	0.768660i	$0.279076\pi$
$450$	0	0
$451$	633752.	0.146716
$452$	3.85586e6i	0.887719i
$453$	1.80669e6i	0.413654i
$454$	4.14609e6	0.944059
$455$	0	0
$456$	999024.	0.224990
$457$	5.34706e6i	1.19764i	0.800885	+	0.598818i	$0.204363\pi$
$457$	5.34706e6i	1.19764i	−0.800885	+	0.598818i	$0.795637\pi$
$458$	9.25198e6i	2.06097i
$459$	7.03924e6	1.55953
$460$	0	0
$461$	3.14978e6	0.690284	0.345142	−	0.938550i	$-0.387831\pi$
$461$	3.14978e6	0.690284	0.345142	+	0.938550i	$0.387831\pi$
$462$	− 3.00188e7i	− 6.54318i
$463$	− 5.22334e6i	− 1.13239i	−0.824272	−	0.566194i	$-0.808415\pi$
$463$	− 5.22334e6i	− 1.13239i	0.824272	−	0.566194i	$-0.191585\pi$
$464$	−1.33378e6	−0.287599
$465$	0	0
$466$	−4.97206e6	−1.06065
$467$	− 6.64502e6i	− 1.40995i	−0.709231	−	0.704976i	$-0.750958\pi$
$467$	− 6.64502e6i	− 1.40995i	0.709231	−	0.704976i	$-0.249042\pi$
$468$	6.09999e6i	1.28740i
$469$	−6.28681e6	−1.31977
$470$	0	0
$471$	−7.97561e6	−1.65658
$472$	1.32617e7i	2.73995i
$473$	1.96482e6i	0.403804i
$474$	−2.08619e7	−4.26489
$475$	0	0
$476$	7.88963e6	1.59602
$477$	9.06300e6i	1.82379i
$478$	− 5.51590e6i	− 1.10420i
$479$	−6.76505e6	−1.34720	−0.673600	−	0.739096i	$-0.735253\pi$
$479$	−6.76505e6	−1.34720	−0.673600	+	0.739096i	$0.735253\pi$
$480$	0	0
$481$	1.17007e6	0.230595
$482$	9.30129e6i	1.82358i
$483$	− 1.91703e7i	− 3.73905i
$484$	−9.69962e6	−1.88209
$485$	0	0
$486$	1.84262e7	3.53871
$487$	− 1.34500e6i	− 0.256980i	−0.991711	−	0.128490i	$-0.958987\pi$
$487$	− 1.34500e6i	− 0.256980i	0.991711	−	0.128490i	$-0.0410130\pi$
$488$	− 3.84307e6i	− 0.730515i
$489$	1.53983e6	0.291207
$490$	0	0
$491$	−7.33455e6	−1.37300	−0.686498	−	0.727131i	$-0.740853\pi$
$491$	−7.33455e6	−1.37300	−0.686498	+	0.727131i	$0.740853\pi$
$492$	1.95923e6i	0.364899i
$493$	− 1.31087e6i	− 0.242908i
$494$	−204559.	−0.0377138
$495$	0	0
$496$	4.59627e6	0.838883
$497$	3.99267e6i	0.725057i
$498$	1.44528e7i	2.61143i
$499$	−4.61062e6	−0.828912	−0.414456	−	0.910069i	$-0.636028\pi$
$499$	−4.61062e6	−0.828912	−0.414456	+	0.910069i	$0.636028\pi$
$500$	0	0
$501$	1.19557e7	2.12805
$502$	1.45711e7i	2.58068i
$503$	7.67903e6i	1.35328i	0.736316	+	0.676638i	$0.236564\pi$
$503$	7.67903e6i	1.35328i	−0.736316	+	0.676638i	$0.763436\pi$
$504$	3.10563e7	5.44594
$505$	0	0
$506$	−1.89832e7	−3.29604
$507$	− 827673.i	− 0.143001i
$508$	− 1.13658e6i	− 0.195407i
$509$	−3.69348e6	−0.631890	−0.315945	−	0.948778i	$-0.602322\pi$
$509$	−3.69348e6	−0.631890	−0.315945	+	0.948778i	$0.602322\pi$
$510$	0	0
$511$	1.58545e7	2.68596
$512$	7.55257e6i	1.27327i
$513$	1.29043e6i	0.216492i
$514$	−3.92592e6	−0.655441
$515$	0	0
$516$	−6.07421e6	−1.00430
$517$	4.39212e6i	0.722683i
$518$	− 1.26501e7i	− 2.07143i
$519$	1.69310e7	2.75908
$520$	0	0
$521$	1.07885e7	1.74127	0.870636	−	0.491927i	$-0.163707\pi$
$521$	1.07885e7	1.74127	0.870636	+	0.491927i	$0.163707\pi$
$522$	− 1.09576e7i	− 1.76010i
$523$	7.52060e6i	1.20226i	0.799151	+	0.601130i	$0.205283\pi$
$523$	7.52060e6i	1.20226i	−0.799151	+	0.601130i	$0.794717\pi$
$524$	1.10019e6	0.175041
$525$	0	0
$526$	−1.19619e7	−1.88511
$527$	4.51734e6i	0.708526i
$528$	− 1.14774e7i	− 1.79167i
$529$	−5.68650e6	−0.883498
$530$	0	0
$531$	−2.88957e7	−4.44731
$532$	1.44633e6i	0.221558i
$533$	− 188915.i	− 0.0288037i
$534$	−3.41617e7	−5.18426
$535$	0	0
$536$	−9.06305e6	−1.36258
$537$	577226.i	0.0863794i
$538$	− 9.33793e6i	− 1.39090i
$539$	1.09369e7	1.62152
$540$	0	0
$541$	−5.06833e6	−0.744512	−0.372256	−	0.928130i	$-0.621416\pi$
$541$	−5.06833e6	−0.744512	−0.372256	+	0.928130i	$0.621416\pi$
$542$	− 6.22358e6i	− 0.910002i
$543$	1.69705e7i	2.46999i
$544$	−1.40517e6	−0.203579
$545$	0	0
$546$	−8.94830e6	−1.28457
$547$	− 6.41454e6i	− 0.916636i	−0.888788	−	0.458318i	$-0.848452\pi$
$547$	− 6.41454e6i	− 0.916636i	0.888788	−	0.458318i	$-0.151548\pi$
$548$	6.42238e6i	0.913576i
$549$	8.37364e6	1.18572
$550$	0	0
$551$	240309.	0.0337202
$552$	− 2.76359e7i	− 3.86034i
$553$	− 1.42227e7i	− 1.97774i
$554$	−6.58384e6	−0.911391
$555$	0	0
$556$	2.05298e6	0.281642
$557$	4.77336e6i	0.651908i	0.945386	+	0.325954i	$0.105685\pi$
$557$	4.77336e6i	0.651908i	−0.945386	+	0.325954i	$0.894315\pi$
$558$	3.77604e7i	5.13394i
$559$	585693.	0.0792758
$560$	0	0
$561$	1.12803e7	1.51326
$562$	2.00121e7i	2.67271i
$563$	− 4.94279e6i	− 0.657205i	−0.944468	−	0.328603i	$-0.893422\pi$
$563$	− 4.94279e6i	− 0.657205i	0.944468	−	0.328603i	$-0.106578\pi$
$564$	−1.35781e7	−1.79739
$565$	0	0
$566$	2.01536e7	2.64430
$567$	2.88962e7i	3.77471i
$568$	5.75582e6i	0.748577i
$569$	−1.01177e7	−1.31008	−0.655042	−	0.755592i	$-0.727349\pi$
$569$	−1.01177e7	−1.31008	−0.655042	+	0.755592i	$0.727349\pi$
$570$	0	0
$571$	−4.01134e6	−0.514872	−0.257436	−	0.966295i	$-0.582878\pi$
$571$	−4.01134e6	−0.514872	−0.257436	+	0.966295i	$0.582878\pi$
$572$	5.79492e6i	0.740555i
$573$	− 7.45485e6i	− 0.948533i
$574$	−2.04243e6	−0.258743
$575$	0	0
$576$	−2.50869e7	−3.15058
$577$	− 715280.i	− 0.0894410i	−0.999000	−	0.0447205i	$-0.985760\pi$
$577$	− 715280.i	− 0.0894410i	0.999000	−	0.0447205i	$-0.0142397\pi$
$578$	− 9.12105e6i	− 1.13560i
$579$	−2.11349e7	−2.62002
$580$	0	0
$581$	−9.85327e6	−1.21099
$582$	− 5.38008e6i	− 0.658386i
$583$	8.60974e6i	1.04910i
$584$	2.28558e7	2.77309
$585$	0	0
$586$	1.64107e6	0.197417
$587$	− 7.23624e6i	− 0.866798i	−0.901202	−	0.433399i	$-0.857314\pi$
$587$	− 7.23624e6i	− 0.866798i	0.901202	−	0.433399i	$-0.142686\pi$
$588$	3.38112e7i	4.03290i
$589$	−828119.	−0.0983568
$590$	0	0
$591$	1.77982e7	2.09608
$592$	− 4.83663e6i	− 0.567203i
$593$	− 9.23201e6i	− 1.07810i	−0.842274	−	0.539050i	$-0.818783\pi$
$593$	− 9.23201e6i	− 1.07810i	0.842274	−	0.539050i	$-0.181217\pi$
$594$	5.58982e7	6.50028
$595$	0	0
$596$	1.50399e7	1.73432
$597$	63435.0i	0.00728439i
$598$	5.65869e6i	0.647087i
$599$	1.13007e7	1.28688	0.643441	−	0.765495i	$-0.277506\pi$
$599$	1.13007e7	1.28688	0.643441	+	0.765495i	$0.277506\pi$
$600$	0	0
$601$	−1.57883e6	−0.178299	−0.0891494	−	0.996018i	$-0.528415\pi$
$601$	−1.57883e6	−0.178299	−0.0891494	+	0.996018i	$0.528415\pi$
$602$	− 6.33216e6i	− 0.712132i
$603$	− 1.97474e7i	− 2.21165i
$604$	−3.77067e6	−0.420558
$605$	0	0
$606$	−2.31927e7	−2.56549
$607$	− 1.76228e7i	− 1.94135i	−0.240405	−	0.970673i	$-0.577280\pi$
$607$	− 1.76228e7i	− 1.94135i	0.240405	−	0.970673i	$-0.422720\pi$
$608$	− 257596.i	− 0.0282606i
$609$	1.05122e7	1.14855
$610$	0	0
$611$	1.30925e6	0.141879
$612$	2.47820e7i	2.67460i
$613$	985236.i	0.105898i	0.998597	+	0.0529492i	$0.0168621\pi$
$613$	985236.i	0.105898i	−0.998597	+	0.0529492i	$0.983138\pi$
$614$	406546.	0.0435200
$615$	0	0
$616$	2.95031e7	3.13268
$617$	− 1.09020e7i	− 1.15290i	−0.817132	−	0.576451i	$-0.804437\pi$
$617$	− 1.09020e7i	− 1.15290i	0.817132	−	0.576451i	$-0.195563\pi$
$618$	3.05239e7i	3.21491i
$619$	−1.39085e7	−1.45900	−0.729499	−	0.683982i	$-0.760247\pi$
$619$	−1.39085e7	−1.45900	−0.729499	+	0.683982i	$0.760247\pi$
$620$	0	0
$621$	3.56971e7	3.71453
$622$	1.29882e7i	1.34608i
$623$	− 2.32900e7i	− 2.40408i
$624$	−3.42129e6	−0.351745
$625$	0	0
$626$	2.14357e7	2.18626
$627$	2.06790e6i	0.210068i
$628$	− 1.66456e7i	− 1.68422i
$629$	4.75357e6	0.479063
$630$	0	0
$631$	−3.18595e6	−0.318541	−0.159271	−	0.987235i	$-0.550914\pi$
$631$	−3.18595e6	−0.318541	−0.159271	+	0.987235i	$0.550914\pi$
$632$	− 2.05035e7i	− 2.04190i
$633$	2.77834e6i	0.275598i
$634$	3.12445e7	3.08710
$635$	0	0
$636$	−2.66168e7	−2.60924
$637$	− 3.26018e6i	− 0.318341i
$638$	− 1.04095e7i	− 1.01247i
$639$	−1.25413e7	−1.21504
$640$	0	0
$641$	−9.63334e6	−0.926044	−0.463022	−	0.886347i	$-0.653235\pi$
$641$	−9.63334e6	−0.926044	−0.463022	+	0.886347i	$0.653235\pi$
$642$	1.81622e7i	1.73912i
$643$	1.78705e7i	1.70455i	0.523093	+	0.852276i	$0.324778\pi$
$643$	1.78705e7i	1.70455i	−0.523093	+	0.852276i	$0.675222\pi$
$644$	4.00096e7	3.80146
$645$	0	0
$646$	−831047.	−0.0783510
$647$	− 1.14547e6i	− 0.107578i	−0.998552	−	0.0537891i	$-0.982870\pi$
$647$	− 1.14547e6i	− 0.107578i	0.998552	−	0.0537891i	$-0.0171299\pi$
$648$	4.16567e7i	3.89716i
$649$	−2.74506e7	−2.55823
$650$	0	0
$651$	−3.62256e7	−3.35014
$652$	3.21372e6i	0.296067i
$653$	− 1.04972e7i	− 0.963366i	−0.876345	−	0.481683i	$-0.840026\pi$
$653$	− 1.04972e7i	− 0.963366i	0.876345	−	0.481683i	$-0.159974\pi$
$654$	5.01666e7	4.58639
$655$	0	0
$656$	−780903.	−0.0708496
$657$	4.98002e7i	4.50109i
$658$	− 1.41548e7i	− 1.27449i
$659$	1.21009e7	1.08543	0.542717	−	0.839915i	$-0.317395\pi$
$659$	1.21009e7	1.08543	0.542717	+	0.839915i	$0.317395\pi$
$660$	0	0
$661$	−6.81298e6	−0.606504	−0.303252	−	0.952910i	$-0.598072\pi$
$661$	−6.81298e6	−0.606504	−0.303252	+	0.952910i	$0.598072\pi$
$662$	− 1.74186e7i	− 1.54479i
$663$	− 3.36253e6i	− 0.297086i
$664$	−1.42045e7	−1.25027
$665$	0	0
$666$	3.97350e7	3.47127
$667$	− 6.64764e6i	− 0.578566i
$668$	2.49523e7i	2.16357i
$669$	−2.38352e7	−2.05899
$670$	0	0
$671$	7.95486e6	0.682066
$672$	− 1.12684e7i	− 0.962585i
$673$	− 9.41818e6i	− 0.801548i	−0.916177	−	0.400774i	$-0.868741\pi$
$673$	− 9.41818e6i	− 0.801548i	0.916177	−	0.400774i	$-0.131259\pi$
$674$	1.37631e7	1.16699
$675$	0	0
$676$	1.72741e6	0.145388
$677$	− 2.81424e6i	− 0.235988i	−0.993014	−	0.117994i	$-0.962354\pi$
$677$	− 2.81424e6i	− 0.235988i	0.993014	−	0.117994i	$-0.0376463\pi$
$678$	− 1.77670e7i	− 1.48436i
$679$	3.66790e6	0.305312
$680$	0	0
$681$	−1.24939e7	−1.03236
$682$	3.58719e7i	2.95321i
$683$	1.31222e6i	0.107635i	0.998551	+	0.0538176i	$0.0171389\pi$
$683$	1.31222e6i	0.107635i	−0.998551	+	0.0538176i	$0.982861\pi$
$684$	−4.54304e6	−0.371284
$685$	0	0
$686$	−4.53860e6	−0.368224
$687$	− 2.78800e7i	− 2.25373i
$688$	− 2.42103e6i	− 0.194998i
$689$	2.56647e6	0.205963
$690$	0	0
$691$	−8.31043e6	−0.662107	−0.331054	−	0.943612i	$-0.607404\pi$
$691$	−8.31043e6	−0.662107	−0.331054	+	0.943612i	$0.607404\pi$
$692$	3.53361e7i	2.80513i
$693$	6.42840e7i	5.08476i
$694$	−2.78117e7	−2.19194
$695$	0	0
$696$	1.51543e7	1.18581
$697$	− 767492.i	− 0.0598401i
$698$	− 3.17191e7i	− 2.46424i
$699$	1.49829e7	1.15985
$700$	0	0
$701$	−1.04904e7	−0.806298	−0.403149	−	0.915134i	$-0.632084\pi$
$701$	−1.04904e7	−0.806298	−0.403149	+	0.915134i	$0.632084\pi$
$702$	− 1.66627e7i	− 1.27615i
$703$	871424.i	0.0665030i
$704$	−2.38322e7	−1.81231
$705$	0	0
$706$	2.76316e7	2.08639
$707$	− 1.58118e7i	− 1.18969i
$708$	− 8.48630e7i	− 6.36261i
$709$	−1.69019e7	−1.26276	−0.631380	−	0.775474i	$-0.717511\pi$
$709$	−1.69019e7	−1.26276	−0.631380	+	0.775474i	$0.717511\pi$
$710$	0	0
$711$	4.46748e7	3.31428
$712$	− 3.35748e7i	− 2.48207i
$713$	2.29082e7i	1.68759i
$714$	−3.63537e7	−2.66872
$715$	0	0
$716$	−1.20471e6	−0.0878211
$717$	1.66217e7i	1.20747i
$718$	− 6.30003e6i	− 0.456070i
$719$	4.94815e6	0.356961	0.178481	−	0.983943i	$-0.442882\pi$
$719$	4.94815e6	0.356961	0.178481	+	0.983943i	$0.442882\pi$
$720$	0	0
$721$	−2.08099e7	−1.49084
$722$	2.36596e7i	1.68913i
$723$	− 2.80286e7i	− 1.99414i
$724$	−3.54185e7	−2.51122
$725$	0	0
$726$	4.46937e7	3.14706
$727$	− 1.84144e7i	− 1.29218i	−0.763262	−	0.646089i	$-0.776404\pi$
$727$	− 1.84144e7i	− 1.29218i	0.763262	−	0.646089i	$-0.223596\pi$
$728$	− 8.79456e6i	− 0.615015i
$729$	−1.85679e7	−1.29403
$730$	0	0
$731$	2.37946e6	0.164696
$732$	2.45923e7i	1.69637i
$733$	− 1.37224e7i	− 0.943344i	−0.881774	−	0.471672i	$-0.843651\pi$
$733$	− 1.37224e7i	− 0.943344i	0.881774	−	0.471672i	$-0.156349\pi$
$734$	−3.33273e7	−2.28328
$735$	0	0
$736$	−7.12586e6	−0.484890
$737$	− 1.87598e7i	− 1.27221i
$738$	− 6.41546e6i	− 0.433598i
$739$	3.74000e6	0.251919	0.125960	−	0.992035i	$-0.459799\pi$
$739$	3.74000e6	0.251919	0.125960	+	0.992035i	$0.459799\pi$
$740$	0	0
$741$	616420.	0.0412412
$742$	− 2.77472e7i	− 1.85016i
$743$	8.50626e6i	0.565284i	0.959226	+	0.282642i	$0.0912108\pi$
$743$	8.50626e6i	0.565284i	−0.959226	+	0.282642i	$0.908789\pi$
$744$	−5.22227e7	−3.45881
$745$	0	0
$746$	3.82303e7	2.51513
$747$	− 3.09500e7i	− 2.02936i
$748$	2.35426e7i	1.53851i
$749$	−1.23822e7	−0.806477
$750$	0	0
$751$	−1.43639e7	−0.929335	−0.464667	−	0.885485i	$-0.653826\pi$
$751$	−1.43639e7	−0.929335	−0.464667	+	0.885485i	$0.653826\pi$
$752$	− 5.41192e6i	− 0.348985i
$753$	− 4.39088e7i	− 2.82205i
$754$	−3.10298e6	−0.198770
$755$	0	0
$756$	−1.17813e8	−7.49703
$757$	− 1.37525e7i	− 0.872253i	−0.899885	−	0.436126i	$-0.856350\pi$
$757$	− 1.37525e7i	− 0.872253i	0.899885	−	0.436126i	$-0.143650\pi$
$758$	− 1.80004e6i	− 0.113791i
$759$	5.72041e7	3.60432
$760$	0	0
$761$	−1.54163e7	−0.964978	−0.482489	−	0.875902i	$-0.660267\pi$
$761$	−1.54163e7	−0.964978	−0.482489	+	0.875902i	$0.660267\pi$
$762$	5.23710e6i	0.326741i
$763$	3.42014e7i	2.12683i
$764$	1.55587e7	0.964364
$765$	0	0
$766$	−2.74476e7	−1.69018
$767$	8.18274e6i	0.502239i
$768$	− 5.54253e7i	− 3.39082i
$769$	2.69620e7	1.64413	0.822065	−	0.569393i	$-0.192822\pi$
$769$	2.69620e7	1.64413	0.822065	+	0.569393i	$0.192822\pi$
$770$	0	0
$771$	1.18304e7	0.716743
$772$	− 4.41099e7i	− 2.66375i
$773$	− 2.34249e7i	− 1.41003i	−0.709191	−	0.705017i	$-0.750939\pi$
$773$	− 2.34249e7i	− 1.41003i	0.709191	−	0.705017i	$-0.249061\pi$
$774$	1.98899e7	1.19338
$775$	0	0
$776$	5.28764e6	0.315216
$777$	3.81199e7i	2.26516i
$778$	− 3.40602e7i	− 2.01743i
$779$	140697.	0.00830693
$780$	0	0
$781$	−1.19141e7	−0.698930
$782$	2.29892e7i	1.34433i
$783$	1.95748e7i	1.14102i
$784$	−1.34764e7	−0.783037
$785$	0	0
$786$	−5.06943e6	−0.292686
$787$	− 6.92601e6i	− 0.398608i	−0.979938	−	0.199304i	$-0.936132\pi$
$787$	− 6.92601e6i	− 0.398608i	0.979938	−	0.199304i	$-0.0638682\pi$
$788$	3.71459e7i	2.13106i
$789$	3.60462e7	2.06142
$790$	0	0
$791$	1.21127e7	0.688337
$792$	9.26718e7i	5.24970i
$793$	− 2.37126e6i	− 0.133905i
$794$	1.20044e7	0.675757
$795$	0	0
$796$	−132393.	−0.00740597
$797$	6.71294e6i	0.374341i	0.982327	+	0.187170i	$0.0599316\pi$
$797$	6.71294e6i	0.374341i	−0.982327	+	0.187170i	$0.940068\pi$
$798$	− 6.66436e6i	− 0.370468i
$799$	5.31898e6	0.294755
$800$	0	0
$801$	7.31559e7	4.02873
$802$	2.63615e7i	1.44722i
$803$	4.73096e7i	2.58917i
$804$	5.79955e7	3.16413
$805$	0	0
$806$	1.06931e7	0.579781
$807$	2.81390e7i	1.52099i
$808$	− 2.27943e7i	− 1.22828i
$809$	−1.59763e7	−0.858235	−0.429117	−	0.903249i	$-0.641175\pi$
$809$	−1.59763e7	−0.858235	−0.429117	+	0.903249i	$0.641175\pi$
$810$	0	0
$811$	−2.53698e7	−1.35445	−0.677227	−	0.735774i	$-0.736819\pi$
$811$	−2.53698e7	−1.35445	−0.677227	+	0.735774i	$0.736819\pi$
$812$	2.19395e7i	1.16772i
$813$	1.87542e7i	0.995114i
$814$	3.77478e7	1.99678
$815$	0	0
$816$	−1.38994e7	−0.730755
$817$	436202.i	0.0228630i
$818$	5.90221e7i	3.08412i
$819$	1.91624e7	0.998252
$820$	0	0
$821$	1.23967e7	0.641874	0.320937	−	0.947101i	$-0.396002\pi$
$821$	1.23967e7	0.641874	0.320937	+	0.947101i	$0.396002\pi$
$822$	− 2.95929e7i	− 1.52759i
$823$	− 1.23249e7i	− 0.634286i	−0.948378	−	0.317143i	$-0.897277\pi$
$823$	− 1.23249e7i	− 0.634286i	0.948378	−	0.317143i	$-0.102723\pi$
$824$	−2.99995e7	−1.53920
$825$	0	0
$826$	8.84669e7	4.51160
$827$	2.17696e6i	0.110684i	0.998467	+	0.0553422i	$0.0176250\pi$
$827$	2.17696e6i	0.110684i	−0.998467	+	0.0553422i	$0.982375\pi$
$828$	1.25674e8i	6.37043i
$829$	948009.	0.0479100	0.0239550	−	0.999713i	$-0.492374\pi$
$829$	948009.	0.0479100	0.0239550	+	0.999713i	$0.492374\pi$
$830$	0	0
$831$	1.98398e7	0.996633
$832$	7.10414e6i	0.355798i
$833$	− 1.32449e7i	− 0.661358i
$834$	−9.45966e6	−0.470934
$835$	0	0
$836$	−4.31584e6	−0.213575
$837$	− 6.74558e7i	− 3.32817i
$838$	1.88388e7i	0.926711i
$839$	−2.05015e7	−1.00550	−0.502748	−	0.864433i	$-0.667678\pi$
$839$	−2.05015e7	−1.00550	−0.502748	+	0.864433i	$0.667678\pi$
$840$	0	0
$841$	−1.68659e7	−0.822278
$842$	− 5.06444e7i	− 2.46179i
$843$	− 6.03046e7i	− 2.92268i
$844$	−5.79856e6	−0.280198
$845$	0	0
$846$	4.44614e7	2.13578
$847$	3.04702e7i	1.45938i
$848$	− 1.06088e7i	− 0.506615i
$849$	−6.07310e7	−2.89162
$850$	0	0
$851$	2.41061e7	1.14105
$852$	− 3.68322e7i	− 1.73832i
$853$	3.75405e7i	1.76655i	0.468851	+	0.883277i	$0.344668\pi$
$853$	3.75405e7i	1.76655i	−0.468851	+	0.883277i	$0.655332\pi$
$854$	−2.56366e7	−1.20286
$855$	0	0
$856$	−1.78501e7	−0.832639
$857$	− 2.62833e7i	− 1.22244i	−0.791460	−	0.611221i	$-0.790678\pi$
$857$	− 2.62833e7i	− 1.22244i	0.791460	−	0.611221i	$-0.209322\pi$
$858$	− 2.67017e7i	− 1.23828i
$859$	1.04479e7	0.483110	0.241555	−	0.970387i	$-0.422343\pi$
$859$	1.04479e7	0.483110	0.241555	+	0.970387i	$0.422343\pi$
$860$	0	0
$861$	6.15470e6	0.282943
$862$	3.10944e7i	1.42532i
$863$	− 8.76214e6i	− 0.400482i	−0.979747	−	0.200241i	$-0.935827\pi$
$863$	− 8.76214e6i	− 0.400482i	0.979747	−	0.200241i	$-0.0641726\pi$
$864$	2.09829e7	0.956274
$865$	0	0
$866$	−3.36365e7	−1.52411
$867$	2.74855e7i	1.24181i
$868$	− 7.56050e7i	− 3.40605i
$869$	4.24406e7	1.90648
$870$	0	0
$871$	−5.59210e6	−0.249764
$872$	4.93047e7i	2.19582i
$873$	1.15212e7i	0.511637i
$874$	−4.21438e6	−0.186619
$875$	0	0
$876$	−1.46257e8	−6.43955
$877$	1.51823e7i	0.666558i	0.942828	+	0.333279i	$0.108155\pi$
$877$	1.51823e7i	0.666558i	−0.942828	+	0.333279i	$0.891845\pi$
$878$	− 282772.i	− 0.0123794i
$879$	−4.94523e6	−0.215881
$880$	0	0
$881$	2.86755e7	1.24472	0.622359	−	0.782732i	$-0.286175\pi$
$881$	2.86755e7	1.24472	0.622359	+	0.782732i	$0.286175\pi$
$882$	− 1.10714e8i	− 4.79217i
$883$	− 1.89214e6i	− 0.0816681i	−0.999166	−	0.0408340i	$-0.986999\pi$
$883$	− 1.89214e6i	− 0.0816681i	0.999166	−	0.0408340i	$-0.0130015\pi$
$884$	7.01781e6	0.302045
$885$	0	0
$886$	5.24938e6	0.224659
$887$	− 1.08388e7i	− 0.462564i	−0.972887	−	0.231282i	$-0.925708\pi$
$887$	− 1.08388e7i	− 0.462564i	0.972887	−	0.231282i	$-0.0742920\pi$
$888$	5.49536e7i	2.33864i
$889$	−3.57043e6	−0.151519
$890$	0	0
$891$	−8.62262e7	−3.63869
$892$	− 4.97456e7i	− 2.09335i
$893$	975077.i	0.0409176i
$894$	−6.93006e7	−2.89997
$895$	0	0
$896$	6.43626e7	2.67833
$897$	− 1.70520e7i	− 0.707609i
$898$	− 5.25558e7i	− 2.17485i
$899$	−1.25618e7	−0.518387
$900$	0	0
$901$	1.04266e7	0.427890
$902$	− 6.09462e6i	− 0.249419i
$903$	1.90814e7i	0.778737i
$904$	1.74617e7	0.710666
$905$	0	0
$906$	1.73744e7	0.703217
$907$	2.32175e7i	0.937123i	0.883431	+	0.468561i	$0.155227\pi$
$907$	2.32175e7i	0.937123i	−0.883431	+	0.468561i	$0.844773\pi$
$908$	− 2.60755e7i	− 1.04959i
$909$	4.96662e7	1.99366
$910$	0	0
$911$	−2.42940e6	−0.0969849	−0.0484924	−	0.998824i	$-0.515442\pi$
$911$	−2.42940e6	−0.0969849	−0.0484924	+	0.998824i	$0.515442\pi$
$912$	− 2.54805e6i	− 0.101443i
$913$	− 2.94021e7i	− 1.16735i
$914$	5.14212e7	2.03599
$915$	0	0
$916$	5.81874e7	2.29134
$917$	− 3.45612e6i	− 0.135727i
$918$	− 6.76943e7i	− 2.65122i
$919$	3.69038e7	1.44139	0.720697	−	0.693251i	$-0.243822\pi$
$919$	3.69038e7	1.44139	0.720697	+	0.693251i	$0.243822\pi$
$920$	0	0
$921$	−1.22509e6	−0.0475904
$922$	− 3.02906e7i	− 1.17349i
$923$	3.55147e6i	0.137216i
$924$	−1.88794e8	−7.27458
$925$	0	0
$926$	−5.02314e7	−1.92507
$927$	− 6.53657e7i	− 2.49833i
$928$	− 3.90751e6i	− 0.148947i
$929$	−2.16356e7	−0.822489	−0.411245	−	0.911525i	$-0.634906\pi$
$929$	−2.16356e7	−0.822489	−0.411245	+	0.911525i	$0.634906\pi$
$930$	0	0
$931$	2.42806e6	0.0918090
$932$	3.12702e7i	1.17921i
$933$	− 3.91387e7i	− 1.47198i
$934$	−6.39033e7	−2.39693
$935$	0	0
$936$	2.76245e7	1.03063
$937$	1.41082e7i	0.524956i	0.964938	+	0.262478i	$0.0845398\pi$
$937$	1.41082e7i	0.524956i	−0.964938	+	0.262478i	$0.915460\pi$
$938$	6.04584e7i	2.24362i
$939$	−6.45945e7	−2.39074
$940$	0	0
$941$	−4.39029e7	−1.61629	−0.808145	−	0.588983i	$-0.799528\pi$
$941$	−4.39029e7	−1.61629	−0.808145	+	0.588983i	$0.799528\pi$
$942$	7.66992e7i	2.81620i
$943$	− 3.89208e6i	− 0.142529i
$944$	3.38244e7	1.23538
$945$	0	0
$946$	1.88951e7	0.686471
$947$	− 4.04056e6i	− 0.146409i	−0.997317	−	0.0732044i	$-0.976677\pi$
$947$	− 4.04056e6i	− 0.146409i	0.997317	−	0.0732044i	$-0.0233225\pi$
$948$	1.31204e8i	4.74162i
$949$	1.41025e7	0.508312
$950$	0	0
$951$	−9.41526e7	−3.37583
$952$	− 3.57291e7i	− 1.27770i
$953$	2.21804e7i	0.791110i	0.918442	+	0.395555i	$0.129448\pi$
$953$	2.21804e7i	0.791110i	−0.918442	+	0.395555i	$0.870552\pi$
$954$	8.71563e7	3.10047
$955$	0	0
$956$	−3.46905e7	−1.22762
$957$	3.13683e7i	1.10716i
$958$	6.50576e7i	2.29026i
$959$	2.01752e7	0.708387
$960$	0	0
$961$	1.46597e7	0.512055
$962$	− 1.12522e7i	− 0.392013i
$963$	− 3.88935e7i	− 1.35148i
$964$	5.84975e7	2.02742
$965$	0	0
$966$	−1.84356e8	−6.35643
$967$	− 2.23431e7i	− 0.768383i	−0.923253	−	0.384191i	$-0.874480\pi$
$967$	− 2.23431e7i	− 0.768383i	0.923253	−	0.384191i	$-0.125520\pi$
$968$	4.39258e7i	1.50672i
$969$	2.50429e6	0.0856791
$970$	0	0
$971$	−5.29498e7	−1.80226	−0.901128	−	0.433554i	$-0.857259\pi$
$971$	−5.29498e7	−1.80226	−0.901128	+	0.433554i	$0.857259\pi$
$972$	− 1.15885e8i	− 3.93426i
$973$	− 6.44919e6i	− 0.218385i
$974$	−1.29345e7	−0.436869
$975$	0	0
$976$	−9.80190e6	−0.329371
$977$	− 1.28543e7i	− 0.430836i	−0.976522	−	0.215418i	$-0.930889\pi$
$977$	− 1.28543e7i	− 0.430836i	0.976522	−	0.215418i	$-0.0691113\pi$
$978$	− 1.48081e7i	− 0.495054i
$979$	6.94972e7	2.31745
$980$	0	0
$981$	−1.07430e8	−3.56412
$982$	7.05343e7i	2.33411i
$983$	− 2.18478e7i	− 0.721149i	−0.932730	−	0.360574i	$-0.882581\pi$
$983$	− 2.18478e7i	− 0.721149i	0.932730	−	0.360574i	$-0.117419\pi$
$984$	8.87260e6	0.292121
$985$	0	0
$986$	−1.26063e7	−0.412947
$987$	4.26541e7i	1.39370i
$988$	1.28651e6i	0.0419295i
$989$	1.20666e7	0.392279
$990$	0	0
$991$	1.64896e7	0.533368	0.266684	−	0.963784i	$-0.414072\pi$
$991$	1.64896e7	0.533368	0.266684	+	0.963784i	$0.414072\pi$
$992$	1.34655e7i	0.434454i
$993$	5.24896e7i	1.68927i
$994$	3.83964e7	1.23261
$995$	0	0
$996$	9.08961e7	2.90333
$997$	− 5.04197e7i	− 1.60643i	−0.595688	−	0.803216i	$-0.703121\pi$
$997$	− 5.04197e7i	− 1.60643i	0.595688	−	0.803216i	$-0.296879\pi$
$998$	4.43391e7i	1.40916i
$999$	−7.09833e7	−2.25031

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.g.274.2		12
5.2	odd	4		65.6.a.d.1.6	✓	6
5.3	odd	4		325.6.a.g.1.1		6
5.4	even	2	inner	325.6.b.g.274.11		12
15.2	even	4		585.6.a.m.1.1		6
20.7	even	4		1040.6.a.q.1.1		6
65.12	odd	4		845.6.a.h.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.d.1.6	✓	6	5.2	odd	4
325.6.a.g.1.1		6	5.3	odd	4
325.6.b.g.274.2		12	1.1	even	1	trivial
325.6.b.g.274.11		12	5.4	even	2	inner
585.6.a.m.1.1		6	15.2	even	4
845.6.a.h.1.1		6	65.12	odd	4
1040.6.a.q.1.1		6	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-9.61672i q^{2} +28.9791i q^{3} -60.4813 q^{4} +278.684 q^{6} +189.995i q^{7} +273.896i q^{8} -596.790 q^{9} -566.944 q^{11} -1752.69i q^{12} +169.000i q^{13} +1827.13 q^{14} +698.582 q^{16} +686.585i q^{17} +5739.16i q^{18} -125.865 q^{19} -5505.88 q^{21} +5452.14i q^{22} +3481.79i q^{23} -7937.27 q^{24} +1625.23 q^{26} -10252.5i q^{27} -11491.1i q^{28} -1909.26 q^{29} +6579.43 q^{31} +2046.61i q^{32} -16429.5i q^{33} +6602.70 q^{34} +36094.6 q^{36} -6923.49i q^{37} +1210.41i q^{38} -4897.47 q^{39} -1117.84 q^{41} +52948.5i q^{42} -3465.64i q^{43} +34289.5 q^{44} +33483.4 q^{46} -7747.01i q^{47} +20244.3i q^{48} -19291.0 q^{49} -19896.6 q^{51} -10221.3i q^{52} -15186.2i q^{53} -98595.7 q^{54} -52038.8 q^{56} -3647.45i q^{57} +18360.8i q^{58} +48418.6 q^{59} -14031.1 q^{61} -63272.5i q^{62} -113387. i q^{63} +42036.3 q^{64} -157998. q^{66} +33089.4i q^{67} -41525.5i q^{68} -100899. q^{69} +21014.6 q^{71} -163459. i q^{72} -83446.8i q^{73} -66581.3 q^{74} +7612.47 q^{76} -107716. i q^{77} +47097.6i q^{78} -74858.5 q^{79} +152089. q^{81} +10750.0i q^{82} +51860.8i q^{83} +333003. q^{84} -33328.1 q^{86} -55328.7i q^{87} -155284. i q^{88} -122582. q^{89} -32109.1 q^{91} -210583. i q^{92} +190666. i q^{93} -74500.8 q^{94} -59309.0 q^{96} -19305.3i q^{97} +185516. i q^{98} +338346. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 268 q^{4} + 636 q^{6} - 1036 q^{9} - 340 q^{11} + 2880 q^{14} + 7012 q^{16} - 2436 q^{19} - 792 q^{21} - 25236 q^{24} - 16728 q^{29} + 5724 q^{31} + 42968 q^{34} - 4276 q^{36} - 12844 q^{39} + 4496 q^{41}+ \cdots + 64540 q^{99}+O(q^{100}) \) 12 * q - 268 * q^4 + 636 * q^6 - 1036 * q^9 - 340 * q^11 + 2880 * q^14 + 7012 * q^16 - 2436 * q^19 - 792 * q^21 - 25236 * q^24 - 16728 * q^29 + 5724 * q^31 + 42968 * q^34 - 4276 * q^36 - 12844 * q^39 + 4496 * q^41 + 146964 * q^44 - 79772 * q^46 - 171804 * q^49 - 110056 * q^51 - 91008 * q^54 - 377600 * q^56 + 57748 * q^59 + 1944 * q^61 - 28044 * q^64 - 316080 * q^66 - 306176 * q^69 - 148348 * q^71 - 452224 * q^74 - 186844 * q^76 - 429016 * q^79 + 232012 * q^81 + 470536 * q^84 - 609332 * q^86 - 188056 * q^89 + 74360 * q^91 + 251840 * q^94 - 771716 * q^96 + 64540 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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