LMFDB - Embedded newform 325.6.b.g.274.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [325,6,Mod(274,325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [12,0,0,-268,0,636] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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			274.8
Root			$2.75663i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.6.b.g.274.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.75663i q^{2} +23.9621i q^{3} +24.4010 q^{4} -66.0547 q^{6} -85.7300i q^{7} +155.477i q^{8} -331.182 q^{9} +431.046 q^{11} +584.699i q^{12} +169.000i q^{13} +236.326 q^{14} +352.239 q^{16} +438.894i q^{17} -912.946i q^{18} -1617.28 q^{19} +2054.27 q^{21} +1188.23i q^{22} +2173.44i q^{23} -3725.55 q^{24} -465.871 q^{26} -2113.02i q^{27} -2091.90i q^{28} -8289.72 q^{29} -2745.88 q^{31} +5946.25i q^{32} +10328.8i q^{33} -1209.87 q^{34} -8081.16 q^{36} +2137.03i q^{37} -4458.24i q^{38} -4049.59 q^{39} -19520.9 q^{41} +5662.87i q^{42} +8153.29i q^{43} +10517.9 q^{44} -5991.39 q^{46} +13235.5i q^{47} +8440.39i q^{48} +9457.36 q^{49} -10516.8 q^{51} +4123.77i q^{52} +1753.17i q^{53} +5824.83 q^{54} +13329.0 q^{56} -38753.4i q^{57} -22851.7i q^{58} +1976.46 q^{59} +45578.3 q^{61} -7569.39i q^{62} +28392.2i q^{63} -5119.96 q^{64} -28472.6 q^{66} +19457.7i q^{67} +10709.5i q^{68} -52080.3 q^{69} -64224.9 q^{71} -51491.1i q^{72} +1029.22i q^{73} -5891.00 q^{74} -39463.2 q^{76} -36953.6i q^{77} -11163.2i q^{78} +107661. q^{79} -29844.7 q^{81} -53812.1i q^{82} -46473.4i q^{83} +50126.2 q^{84} -22475.6 q^{86} -198639. i q^{87} +67017.6i q^{88} +3410.51 q^{89} +14488.4 q^{91} +53034.2i q^{92} -65797.1i q^{93} -36485.3 q^{94} -142485. q^{96} -133264. i q^{97} +26070.5i q^{98} -142755. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.75663i	0.487308i	0.969862	+	0.243654i	$0.0783462\pi$
$2$	2.75663i	0.487308i	−0.969862	+	0.243654i	$0.921654\pi$
$3$	23.9621i	1.53717i	0.639748	+	0.768584i	$0.279039\pi$
$3$	23.9621i	1.53717i	−0.639748	+	0.768584i	$0.720961\pi$
$4$	24.4010	0.762531
$5$	0	0
$5$	0	0
$6$	−66.0547	−0.749075
$7$	− 85.7300i	− 0.661284i	−0.943756	−	0.330642i	$-0.892735\pi$
$7$	− 85.7300i	− 0.661284i	0.943756	−	0.330642i	$-0.107265\pi$
$8$	155.477i	0.858896i
$9$	−331.182	−1.36289
$10$	0	0
$11$	431.046	1.07409	0.537046	−	0.843553i	$-0.319540\pi$
$11$	431.046	1.07409	0.537046	+	0.843553i	$0.319540\pi$
$12$	584.699i	1.17214i
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	236.326	0.322249
$15$	0	0
$16$	352.239	0.343984
$17$	438.894i	0.368331i	0.982895	+	0.184165i	$0.0589582\pi$
$17$	438.894i	0.368331i	−0.982895	+	0.184165i	$0.941042\pi$
$18$	− 912.946i	− 0.664147i
$19$	−1617.28	−1.02778	−0.513891	−	0.857856i	$-0.671796\pi$
$19$	−1617.28	−1.02778	−0.513891	+	0.857856i	$0.671796\pi$
$20$	0	0
$21$	2054.27	1.01650
$22$	1188.23i	0.523414i
$23$	2173.44i	0.856700i	0.903613	+	0.428350i	$0.140905\pi$
$23$	2173.44i	0.856700i	−0.903613	+	0.428350i	$0.859095\pi$
$24$	−3725.55	−1.32027
$25$	0	0
$26$	−465.871	−0.135155
$27$	− 2113.02i	− 0.557821i
$28$	− 2091.90i	− 0.504249i
$29$	−8289.72	−1.83040	−0.915198	−	0.403005i	$-0.867966\pi$
$29$	−8289.72	−1.83040	−0.915198	+	0.403005i	$0.867966\pi$
$30$	0	0
$31$	−2745.88	−0.513190	−0.256595	−	0.966519i	$-0.582601\pi$
$31$	−2745.88	−0.513190	−0.256595	+	0.966519i	$0.582601\pi$
$32$	5946.25i	1.02652i
$33$	10328.8i	1.65106i
$34$	−1209.87	−0.179491
$35$	0	0
$36$	−8081.16	−1.03924
$37$	2137.03i	0.256629i	0.991734	+	0.128315i	$0.0409567\pi$
$37$	2137.03i	0.256629i	−0.991734	+	0.128315i	$0.959043\pi$
$38$	− 4458.24i	− 0.500846i
$39$	−4049.59	−0.426334
$40$	0	0
$41$	−19520.9	−1.81360	−0.906799	−	0.421562i	$-0.861482\pi$
$41$	−19520.9	−1.81360	−0.906799	+	0.421562i	$0.861482\pi$
$42$	5662.87i	0.495351i
$43$	8153.29i	0.672452i	0.941781	+	0.336226i	$0.109151\pi$
$43$	8153.29i	0.672452i	−0.941781	+	0.336226i	$0.890849\pi$
$44$	10517.9	0.819029
$45$	0	0
$46$	−5991.39	−0.417477
$47$	13235.5i	0.873966i	0.899470	+	0.436983i	$0.143953\pi$
$47$	13235.5i	0.873966i	−0.899470	+	0.436983i	$0.856047\pi$
$48$	8440.39i	0.528761i
$49$	9457.36	0.562704
$50$	0	0
$51$	−10516.8	−0.566186
$52$	4123.77i	0.211488i
$53$	1753.17i	0.0857303i	0.999081	+	0.0428651i	$0.0136486\pi$
$53$	1753.17i	0.0857303i	−0.999081	+	0.0428651i	$0.986351\pi$
$54$	5824.83	0.271831
$55$	0	0
$56$	13329.0	0.567974
$57$	− 38753.4i	− 1.57987i
$58$	− 22851.7i	− 0.891967i
$59$	1976.46	0.0739192	0.0369596	−	0.999317i	$-0.488233\pi$
$59$	1976.46	0.0739192	0.0369596	+	0.999317i	$0.488233\pi$
$60$	0	0
$61$	45578.3	1.56832	0.784158	−	0.620561i	$-0.213095\pi$
$61$	45578.3	1.56832	0.784158	+	0.620561i	$0.213095\pi$
$62$	− 7569.39i	− 0.250082i
$63$	28392.2i	0.901256i
$64$	−5119.96	−0.156249
$65$	0	0
$66$	−28472.6	−0.804576
$67$	19457.7i	0.529546i	0.964311	+	0.264773i	$0.0852970\pi$
$67$	19457.7i	0.529546i	−0.964311	+	0.264773i	$0.914703\pi$
$68$	10709.5i	0.280863i
$69$	−52080.3	−1.31689
$70$	0	0
$71$	−64224.9	−1.51202	−0.756010	−	0.654561i	$-0.772854\pi$
$71$	−64224.9	−1.51202	−0.756010	+	0.654561i	$0.772854\pi$
$72$	− 51491.1i	− 1.17058i
$73$	1029.22i	0.0226048i	0.999936	+	0.0113024i	$0.00359775\pi$
$73$	1029.22i	0.0226048i	−0.999936	+	0.0113024i	$0.996402\pi$
$74$	−5891.00	−0.125058
$75$	0	0
$76$	−39463.2	−0.783715
$77$	− 36953.6i	− 0.710280i
$78$	− 11163.2i	− 0.207756i
$79$	107661.	1.94085	0.970424	−	0.241407i	$-0.0776090\pi$
$79$	107661.	1.94085	0.970424	+	0.241407i	$0.0776090\pi$
$80$	0	0
$81$	−29844.7	−0.505423
$82$	− 53812.1i	− 0.883782i
$83$	− 46473.4i	− 0.740473i	−0.928938	−	0.370237i	$-0.879277\pi$
$83$	− 46473.4i	− 0.740473i	0.928938	−	0.370237i	$-0.120723\pi$
$84$	50126.2	0.775116
$85$	0	0
$86$	−22475.6	−0.327692
$87$	− 198639.i	− 2.81363i
$88$	67017.6i	0.922534i
$89$	3410.51	0.0456398	0.0228199	−	0.999740i	$-0.492736\pi$
$89$	3410.51	0.0456398	0.0228199	+	0.999740i	$0.492736\pi$
$90$	0	0
$91$	14488.4	0.183407
$92$	53034.2i	0.653260i
$93$	− 65797.1i	− 0.788859i
$94$	−36485.3	−0.425891
$95$	0	0
$96$	−142485.	−1.57794
$97$	− 133264.i	− 1.43808i	−0.694967	−	0.719041i	$-0.744581\pi$
$97$	− 133264.i	− 1.43808i	0.694967	−	0.719041i	$-0.255419\pi$
$98$	26070.5i	0.274210i
$99$	−142755.	−1.46387
$100$	0	0
$101$	112628.	1.09860	0.549302	−	0.835624i	$-0.314894\pi$
$101$	112628.	1.09860	0.549302	+	0.835624i	$0.314894\pi$
$102$	− 28991.0i	− 0.275907i
$103$	102456.i	0.951581i	0.879559	+	0.475791i	$0.157838\pi$
$103$	102456.i	0.951581i	−0.879559	+	0.475791i	$0.842162\pi$
$104$	−26275.6	−0.238215
$105$	0	0
$106$	−4832.84	−0.0417771
$107$	90417.9i	0.763475i	0.924271	+	0.381738i	$0.124674\pi$
$107$	90417.9i	0.763475i	−0.924271	+	0.381738i	$0.875326\pi$
$108$	− 51559.8i	− 0.425356i
$109$	−164916.	−1.32953	−0.664764	−	0.747053i	$-0.731468\pi$
$109$	−164916.	−1.32953	−0.664764	+	0.747053i	$0.731468\pi$
$110$	0	0
$111$	−51207.7	−0.394483
$112$	− 30197.5i	− 0.227471i
$113$	− 36453.6i	− 0.268562i	−0.990943	−	0.134281i	$-0.957128\pi$
$113$	− 36453.6i	− 0.268562i	0.990943	−	0.134281i	$-0.0428724\pi$
$114$	106829.	0.769885
$115$	0	0
$116$	−202277.	−1.39573
$117$	− 55969.7i	− 0.377997i
$118$	5448.36i	0.0360214i
$119$	37626.4	0.243571
$120$	0	0
$121$	24749.6	0.153676
$122$	125643.i	0.764253i
$123$	− 467763.i	− 2.78781i
$124$	−67002.3	−0.391323
$125$	0	0
$126$	−78266.9	−0.439189
$127$	− 178579.i	− 0.982476i	−0.871025	−	0.491238i	$-0.836544\pi$
$127$	− 178579.i	− 0.982476i	0.871025	−	0.491238i	$-0.163456\pi$
$128$	176166.i	0.950381i
$129$	−195370.	−1.03367
$130$	0	0
$131$	43555.7	0.221752	0.110876	−	0.993834i	$-0.464634\pi$
$131$	43555.7	0.221752	0.110876	+	0.993834i	$0.464634\pi$
$132$	252032.i	1.25899i
$133$	138649.i	0.679655i
$134$	−53637.6	−0.258052
$135$	0	0
$136$	−68237.9	−0.316358
$137$	347518.i	1.58189i	0.611888	+	0.790944i	$0.290410\pi$
$137$	347518.i	1.58189i	−0.611888	+	0.790944i	$0.709590\pi$
$138$	− 143566.i	− 0.641733i
$139$	−54594.1	−0.239667	−0.119834	−	0.992794i	$-0.538236\pi$
$139$	−54594.1	−0.239667	−0.119834	+	0.992794i	$0.538236\pi$
$140$	0	0
$141$	−317150.	−1.34343
$142$	− 177044.i	− 0.736819i
$143$	72846.8i	0.297900i
$144$	−116655.	−0.468812
$145$	0	0
$146$	−2837.18	−0.0110155
$147$	226618.i	0.864971i
$148$	52145.6i	0.195688i
$149$	249528.	0.920777	0.460388	−	0.887718i	$-0.347710\pi$
$149$	249528.	0.920777	0.460388	+	0.887718i	$0.347710\pi$
$150$	0	0
$151$	398788.	1.42331	0.711655	−	0.702529i	$-0.247946\pi$
$151$	398788.	1.42331	0.711655	+	0.702529i	$0.247946\pi$
$152$	− 251449.i	− 0.882757i
$153$	− 145354.i	− 0.501994i
$154$	101867.	0.346125
$155$	0	0
$156$	−98814.1	−0.325093
$157$	− 432099.i	− 1.39905i	−0.714606	−	0.699527i	$-0.753394\pi$
$157$	− 432099.i	− 1.39905i	0.714606	−	0.699527i	$-0.246606\pi$
$158$	296782.i	0.945791i
$159$	−42009.6	−0.131782
$160$	0	0
$161$	186329.	0.566522
$162$	− 82271.0i	− 0.246297i
$163$	56309.9i	0.166003i	0.996549	+	0.0830014i	$0.0264506\pi$
$163$	56309.9i	0.166003i	−0.996549	+	0.0830014i	$0.973549\pi$
$164$	−476330.	−1.38292
$165$	0	0
$166$	128110.	0.360839
$167$	512611.i	1.42232i	0.703031	+	0.711160i	$0.251830\pi$
$167$	512611.i	1.42232i	−0.703031	+	0.711160i	$0.748170\pi$
$168$	319391.i	0.873072i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	535613.	1.40075
$172$	198948.i	0.512766i
$173$	− 363030.i	− 0.922204i	−0.887347	−	0.461102i	$-0.847454\pi$
$173$	− 363030.i	− 0.922204i	0.887347	−	0.461102i	$-0.152546\pi$
$174$	547575.	1.37110
$175$	0	0
$176$	151831.	0.369471
$177$	47360.0i	0.113626i
$178$	9401.51i	0.0222406i
$179$	812489.	1.89533	0.947665	−	0.319265i	$-0.103436\pi$
$179$	812489.	1.89533	0.947665	+	0.319265i	$0.103436\pi$
$180$	0	0
$181$	−464974.	−1.05495	−0.527475	−	0.849570i	$-0.676861\pi$
$181$	−464974.	−1.05495	−0.527475	+	0.849570i	$0.676861\pi$
$182$	39939.1i	0.0893758i
$183$	1.09215e6i	2.41077i
$184$	−337920.	−0.735816
$185$	0	0
$186$	181378.	0.384418
$187$	189184.i	0.395621i
$188$	322959.i	0.666426i
$189$	−181150.	−0.368878
$190$	0	0
$191$	−15537.6	−0.0308177	−0.0154088	−	0.999881i	$-0.504905\pi$
$191$	−15537.6	−0.0308177	−0.0154088	+	0.999881i	$0.504905\pi$
$192$	− 122685.i	− 0.240181i
$193$	− 319408.i	− 0.617238i	−0.951186	−	0.308619i	$-0.900133\pi$
$193$	− 319408.i	− 0.617238i	0.951186	−	0.308619i	$-0.0998667\pi$
$194$	367360.	0.700789
$195$	0	0
$196$	230769.	0.429079
$197$	− 635878.i	− 1.16737i	−0.811980	−	0.583685i	$-0.801610\pi$
$197$	− 635878.i	− 1.16737i	0.811980	−	0.583685i	$-0.198390\pi$
$198$	− 393522.i	− 0.713355i
$199$	−57426.3	−0.102797	−0.0513983	−	0.998678i	$-0.516368\pi$
$199$	−57426.3	−0.102797	−0.0513983	+	0.998678i	$0.516368\pi$
$200$	0	0
$201$	−466246.	−0.814002
$202$	310473.i	0.535359i
$203$	710678.i	1.21041i
$204$	−256621.	−0.431734
$205$	0	0
$206$	−282434.	−0.463713
$207$	− 719806.i	− 1.16759i
$208$	59528.5i	0.0954039i
$209$	−697121.	−1.10393
$210$	0	0
$211$	−407094.	−0.629489	−0.314745	−	0.949176i	$-0.601919\pi$
$211$	−407094.	−0.629489	−0.314745	+	0.949176i	$0.601919\pi$
$212$	42779.1i	0.0653720i
$213$	− 1.53896e6i	− 2.32423i
$214$	−249249.	−0.372048
$215$	0	0
$216$	328526.	0.479110
$217$	235405.i	0.339364i
$218$	− 454614.i	− 0.647890i
$219$	−24662.3	−0.0347474
$220$	0	0
$221$	−74173.2	−0.102157
$222$	− 141161.i	− 0.192235i
$223$	94338.9i	0.127037i	0.997981	+	0.0635183i	$0.0202321\pi$
$223$	94338.9i	0.127037i	−0.997981	+	0.0635183i	$0.979768\pi$
$224$	509772.	0.678822
$225$	0	0
$226$	100489.	0.130872
$227$	− 251896.i	− 0.324456i	−0.986753	−	0.162228i	$-0.948132\pi$
$227$	− 251896.i	− 0.324456i	0.986753	−	0.162228i	$-0.0518680\pi$
$228$	− 945620.i	− 1.20470i
$229$	428033.	0.539372	0.269686	−	0.962948i	$-0.413080\pi$
$229$	428033.	0.539372	0.269686	+	0.962948i	$0.413080\pi$
$230$	0	0
$231$	885485.	1.09182
$232$	− 1.28886e6i	− 1.57212i
$233$	47973.8i	0.0578914i	0.999581	+	0.0289457i	$0.00921499\pi$
$233$	47973.8i	0.0578914i	−0.999581	+	0.0289457i	$0.990785\pi$
$234$	154288.	0.184201
$235$	0	0
$236$	48227.5	0.0563657
$237$	2.57979e6i	2.98341i
$238$	103722.i	0.118694i
$239$	1.39917e6	1.58444	0.792220	−	0.610236i	$-0.208925\pi$
$239$	1.39917e6	1.58444	0.792220	+	0.610236i	$0.208925\pi$
$240$	0	0
$241$	−576792.	−0.639700	−0.319850	−	0.947468i	$-0.603633\pi$
$241$	−576792.	−0.639700	−0.319850	+	0.947468i	$0.603633\pi$
$242$	68225.6i	0.0748875i
$243$	− 1.22861e6i	− 1.33474i
$244$	1.11216e6	1.19589
$245$	0	0
$246$	1.28945e6	1.35852
$247$	− 273320.i	− 0.285055i
$248$	− 426921.i	− 0.440776i
$249$	1.11360e6	1.13823
$250$	0	0
$251$	676974.	0.678247	0.339124	−	0.940742i	$-0.389870\pi$
$251$	676974.	0.678247	0.339124	+	0.940742i	$0.389870\pi$
$252$	692798.i	0.687235i
$253$	936855.i	0.920176i
$254$	492278.	0.478769
$255$	0	0
$256$	−649464.	−0.619377
$257$	2.05124e6i	1.93724i	0.248545	+	0.968620i	$0.420047\pi$
$257$	2.05124e6i	1.93724i	−0.248545	+	0.968620i	$0.579953\pi$
$258$	− 538562.i	− 0.503717i
$259$	183208.	0.169705
$260$	0	0
$261$	2.74541e6	2.49462
$262$	120067.i	0.108061i
$263$	1.51723e6i	1.35258i	0.736636	+	0.676289i	$0.236413\pi$
$263$	1.51723e6i	1.35258i	−0.736636	+	0.676289i	$0.763587\pi$
$264$	−1.60588e6	−1.41809
$265$	0	0
$266$	−382205.	−0.331201
$267$	81722.8i	0.0701561i
$268$	474786.i	0.403795i
$269$	−180817.	−0.152355	−0.0761777	−	0.997094i	$-0.524272\pi$
$269$	−180817.	−0.152355	−0.0761777	+	0.997094i	$0.524272\pi$
$270$	0	0
$271$	−1.58203e6	−1.30856	−0.654278	−	0.756254i	$-0.727028\pi$
$271$	−1.58203e6	−1.30856	−0.654278	+	0.756254i	$0.727028\pi$
$272$	154596.i	0.126700i
$273$	347172.i	0.281928i
$274$	−957979.	−0.770867
$275$	0	0
$276$	−1.27081e6	−1.00417
$277$	1.53514e6i	1.20212i	0.799202	+	0.601062i	$0.205256\pi$
$277$	1.53514e6i	1.20212i	−0.799202	+	0.601062i	$0.794744\pi$
$278$	− 150496.i	− 0.116792i
$279$	909387.	0.699420
$280$	0	0
$281$	959201.	0.724676	0.362338	−	0.932047i	$-0.381979\pi$
$281$	959201.	0.724676	0.362338	+	0.932047i	$0.381979\pi$
$282$	− 874265.i	− 0.654666i
$283$	2.63997e6i	1.95945i	0.200353	+	0.979724i	$0.435791\pi$
$283$	2.63997e6i	1.95945i	−0.200353	+	0.979724i	$0.564209\pi$
$284$	−1.56715e6	−1.15296
$285$	0	0
$286$	−200812.	−0.145169
$287$	1.67353e6i	1.19930i
$288$	− 1.96929e6i	− 1.39903i
$289$	1.22723e6	0.864333
$290$	0	0
$291$	3.19329e6	2.21058
$292$	25114.0i	0.0172369i
$293$	286709.i	0.195106i	0.995230	+	0.0975532i	$0.0311016\pi$
$293$	286709.i	0.195106i	−0.995230	+	0.0975532i	$0.968898\pi$
$294$	−624703.	−0.421507
$295$	0	0
$296$	−332258.	−0.220418
$297$	− 910810.i	− 0.599152i
$298$	687858.i	0.448702i
$299$	−367312.	−0.237606
$300$	0	0
$301$	698981.	0.444682
$302$	1.09931e6i	0.693591i
$303$	2.69879e6i	1.68874i
$304$	−569669.	−0.353540
$305$	0	0
$306$	400687.	0.244626
$307$	− 1.89896e6i	− 1.14993i	−0.818179	−	0.574963i	$-0.805016\pi$
$307$	− 1.89896e6i	− 1.14993i	0.818179	−	0.574963i	$-0.194984\pi$
$308$	− 901704.i	− 0.541610i
$309$	−2.45507e6	−1.46274
$310$	0	0
$311$	1.74416e6	1.02255	0.511277	−	0.859416i	$-0.329173\pi$
$311$	1.74416e6	1.02255	0.511277	+	0.859416i	$0.329173\pi$
$312$	− 629618.i	− 0.366176i
$313$	605507.i	0.349348i	0.984626	+	0.174674i	$0.0558871\pi$
$313$	605507.i	0.349348i	−0.984626	+	0.174674i	$0.944113\pi$
$314$	1.19114e6	0.681770
$315$	0	0
$316$	2.62704e6	1.47996
$317$	2.19697e6i	1.22793i	0.789332	+	0.613967i	$0.210427\pi$
$317$	2.19697e6i	1.22793i	−0.789332	+	0.613967i	$0.789573\pi$
$318$	− 115805.i	− 0.0642184i
$319$	−3.57325e6	−1.96601
$320$	0	0
$321$	−2.16660e6	−1.17359
$322$	513642.i	0.276071i
$323$	− 709814.i	− 0.378563i
$324$	−728241.	−0.385401
$325$	0	0
$326$	−155226.	−0.0808945
$327$	− 3.95174e6i	− 2.04371i
$328$	− 3.03505e6i	− 1.55769i
$329$	1.13468e6	0.577940
$330$	0	0
$331$	−3.68205e6	−1.84722	−0.923612	−	0.383329i	$-0.874777\pi$
$331$	−3.68205e6	−1.84722	−0.923612	+	0.383329i	$0.874777\pi$
$332$	− 1.13400e6i	− 0.564633i
$333$	− 707745.i	− 0.349757i
$334$	−1.41308e6	−0.693108
$335$	0	0
$336$	723595.	0.349661
$337$	3.46729e6i	1.66309i	0.555460	+	0.831543i	$0.312542\pi$
$337$	3.46729e6i	1.66309i	−0.555460	+	0.831543i	$0.687458\pi$
$338$	− 78732.2i	− 0.0374852i
$339$	873504.	0.412825
$340$	0	0
$341$	−1.18360e6	−0.551214
$342$	1.47649e6i	0.682597i
$343$	− 2.25164e6i	− 1.03339i
$344$	−1.26765e6	−0.577566
$345$	0	0
$346$	1.00074e6	0.449398
$347$	− 3.91532e6i	− 1.74559i	−0.488083	−	0.872797i	$-0.662304\pi$
$347$	− 3.91532e6i	− 1.74559i	0.488083	−	0.872797i	$-0.337696\pi$
$348$	− 4.84699e6i	− 2.14548i
$349$	3.32051e6	1.45929	0.729644	−	0.683827i	$-0.239686\pi$
$349$	3.32051e6	1.45929	0.729644	+	0.683827i	$0.239686\pi$
$350$	0	0
$351$	357101.	0.154712
$352$	2.56311e6i	1.10258i
$353$	3.09575e6i	1.32230i	0.750255	+	0.661148i	$0.229931\pi$
$353$	3.09575e6i	1.32230i	−0.750255	+	0.661148i	$0.770069\pi$
$354$	−130554.	−0.0553710
$355$	0	0
$356$	83219.7	0.0348017
$357$	901608.i	0.374410i
$358$	2.23973e6i	0.923610i
$359$	2.71429e6	1.11153	0.555764	−	0.831340i	$-0.312426\pi$
$359$	2.71429e6	1.11153	0.555764	+	0.831340i	$0.312426\pi$
$360$	0	0
$361$	139488.	0.0563340
$362$	− 1.28176e6i	− 0.514086i
$363$	593053.i	0.236226i
$364$	353531.	0.139854
$365$	0	0
$366$	−3.01066e6	−1.17479
$367$	− 2.54964e6i	− 0.988130i	−0.869425	−	0.494065i	$-0.835511\pi$
$367$	− 2.54964e6i	− 0.988130i	0.869425	−	0.494065i	$-0.164489\pi$
$368$	765573.i	0.294691i
$369$	6.46498e6	2.47173
$370$	0	0
$371$	150299.	0.0566920
$372$	− 1.60551e6i	− 0.601530i
$373$	3.67946e6i	1.36934i	0.728852	+	0.684671i	$0.240054\pi$
$373$	3.67946e6i	1.36934i	−0.728852	+	0.684671i	$0.759946\pi$
$374$	−521510.	−0.192790
$375$	0	0
$376$	−2.05781e6	−0.750646
$377$	− 1.40096e6i	− 0.507660i
$378$	− 499362.i	− 0.179757i
$379$	1.06822e6	0.382000	0.191000	−	0.981590i	$-0.438827\pi$
$379$	1.06822e6	0.382000	0.191000	+	0.981590i	$0.438827\pi$
$380$	0	0
$381$	4.27914e6	1.51023
$382$	− 42831.4i	− 0.0150177i
$383$	− 601784.i	− 0.209625i	−0.994492	−	0.104813i	$-0.966576\pi$
$383$	− 601784.i	− 0.209625i	0.994492	−	0.104813i	$-0.0334243\pi$
$384$	−4.22131e6	−1.46090
$385$	0	0
$386$	880490.	0.300785
$387$	− 2.70022e6i	− 0.916478i
$388$	− 3.25178e6i	− 1.09658i
$389$	−5.00396e6	−1.67664	−0.838319	−	0.545180i	$-0.816461\pi$
$389$	−5.00396e6	−1.67664	−0.838319	+	0.545180i	$0.816461\pi$
$390$	0	0
$391$	−953913.	−0.315549
$392$	1.47040e6i	0.483304i
$393$	1.04369e6i	0.340870i
$394$	1.75288e6	0.568869
$395$	0	0
$396$	−3.48335e6	−1.11625
$397$	− 3.47583e6i	− 1.10683i	−0.832904	−	0.553417i	$-0.813323\pi$
$397$	− 3.47583e6i	− 1.10683i	0.832904	−	0.553417i	$-0.186677\pi$
$398$	− 158303.i	− 0.0500936i
$399$	−3.32233e6	−1.04474
$400$	0	0
$401$	−5.15091e6	−1.59964	−0.799822	−	0.600238i	$-0.795073\pi$
$401$	−5.15091e6	−1.59964	−0.799822	+	0.600238i	$0.795073\pi$
$402$	− 1.28527e6i	− 0.396670i
$403$	− 464054.i	− 0.142333i
$404$	2.74822e6	0.837719
$405$	0	0
$406$	−1.95908e6	−0.589843
$407$	921158.i	0.275644i
$408$	− 1.63512e6i	− 0.486295i
$409$	1.11646e6	0.330015	0.165008	−	0.986292i	$-0.447235\pi$
$409$	1.11646e6	0.330015	0.165008	+	0.986292i	$0.447235\pi$
$410$	0	0
$411$	−8.32726e6	−2.43163
$412$	2.50004e6i	0.725610i
$413$	− 169442.i	− 0.0488816i
$414$	1.98424e6	0.568975
$415$	0	0
$416$	−1.00492e6	−0.284706
$417$	− 1.30819e6i	− 0.368409i
$418$	− 1.92171e6i	− 0.537955i
$419$	−1.79740e6	−0.500160	−0.250080	−	0.968225i	$-0.580457\pi$
$419$	−1.79740e6	−0.500160	−0.250080	+	0.968225i	$0.580457\pi$
$420$	0	0
$421$	5.31302e6	1.46095	0.730477	−	0.682937i	$-0.239298\pi$
$421$	5.31302e6	1.46095	0.730477	+	0.682937i	$0.239298\pi$
$422$	− 1.12221e6i	− 0.306755i
$423$	− 4.38335e6i	− 1.19112i
$424$	−272577.	−0.0736334
$425$	0	0
$426$	4.24235e6	1.13262
$427$	− 3.90743e6i	− 1.03710i
$428$	2.20629e6i	0.582173i
$429$	−1.74556e6	−0.457922
$430$	0	0
$431$	−4.63630e6	−1.20221	−0.601103	−	0.799172i	$-0.705272\pi$
$431$	−4.63630e6	−1.20221	−0.601103	+	0.799172i	$0.705272\pi$
$432$	− 744290.i	− 0.191881i
$433$	− 3.68664e6i	− 0.944955i	−0.881343	−	0.472478i	$-0.843360\pi$
$433$	− 3.68664e6i	− 0.944955i	0.881343	−	0.472478i	$-0.156640\pi$
$434$	−648924.	−0.165375
$435$	0	0
$436$	−4.02412e6	−1.01381
$437$	− 3.51506e6i	− 0.880501i
$438$	− 67984.8i	− 0.0169327i
$439$	1.67032e6	0.413656	0.206828	−	0.978377i	$-0.433686\pi$
$439$	1.67032e6	0.413656	0.206828	+	0.978377i	$0.433686\pi$
$440$	0	0
$441$	−3.13211e6	−0.766903
$442$	− 204468.i	− 0.0497817i
$443$	3.59209e6i	0.869637i	0.900518	+	0.434819i	$0.143187\pi$
$443$	3.59209e6i	0.869637i	−0.900518	+	0.434819i	$0.856813\pi$
$444$	−1.24952e6	−0.300805
$445$	0	0
$446$	−260058.	−0.0619060
$447$	5.97922e6i	1.41539i
$448$	438934.i	0.103325i
$449$	−80455.2	−0.0188338	−0.00941690	−	0.999956i	$-0.502998\pi$
$449$	−80455.2	−0.0188338	−0.00941690	+	0.999956i	$0.502998\pi$
$450$	0	0
$451$	−8.41443e6	−1.94797
$452$	− 889503.i	− 0.204786i
$453$	9.55579e6i	2.18787i
$454$	694384.	0.158110
$455$	0	0
$456$	6.02525e6	1.35695
$457$	− 2.87687e6i	− 0.644362i	−0.946678	−	0.322181i	$-0.895584\pi$
$457$	− 2.87687e6i	− 0.644362i	0.946678	−	0.322181i	$-0.104416\pi$
$458$	1.17993e6i	0.262840i
$459$	927394.	0.205463
$460$	0	0
$461$	81644.7	0.0178927	0.00894635	−	0.999960i	$-0.497152\pi$
$461$	81644.7	0.0178927	0.00894635	+	0.999960i	$0.497152\pi$
$462$	2.44096e6i	0.532053i
$463$	− 2.50214e6i	− 0.542450i	−0.962516	−	0.271225i	$-0.912571\pi$
$463$	− 2.50214e6i	− 0.542450i	0.962516	−	0.271225i	$-0.0874287\pi$
$464$	−2.91997e6	−0.629626
$465$	0	0
$466$	−132246.	−0.0282109
$467$	− 1.38773e6i	− 0.294452i	−0.989103	−	0.147226i	$-0.952966\pi$
$467$	− 1.38773e6i	− 0.294452i	0.989103	−	0.147226i	$-0.0470344\pi$
$468$	− 1.36572e6i	− 0.288235i
$469$	1.66811e6	0.350180
$470$	0	0
$471$	1.03540e7	2.15058
$472$	307293.i	0.0634889i
$473$	3.51444e6i	0.722276i
$474$	−7.11152e6	−1.45384
$475$	0	0
$476$	918122.	0.185730
$477$	− 580618.i	− 0.116841i
$478$	3.85700e6i	0.772111i
$479$	4.96374e6	0.988485	0.494243	−	0.869324i	$-0.335445\pi$
$479$	4.96374e6	0.988485	0.494243	+	0.869324i	$0.335445\pi$
$480$	0	0
$481$	−361158.	−0.0711761
$482$	− 1.59000e6i	− 0.311731i
$483$	4.46484e6i	0.870840i
$484$	603915.	0.117182
$485$	0	0
$486$	3.38682e6	0.650431
$487$	− 8.57371e6i	− 1.63812i	−0.573706	−	0.819061i	$-0.694495\pi$
$487$	− 8.57371e6i	− 1.63812i	0.573706	−	0.819061i	$-0.305505\pi$
$488$	7.08637e6i	1.34702i
$489$	−1.34930e6	−0.255174
$490$	0	0
$491$	2.55302e6	0.477915	0.238958	−	0.971030i	$-0.423194\pi$
$491$	2.55302e6	0.477915	0.238958	+	0.971030i	$0.423194\pi$
$492$	− 1.14139e7i	− 2.12579i
$493$	− 3.63831e6i	− 0.674191i
$494$	753442.	0.138910
$495$	0	0
$496$	−967209.	−0.176529
$497$	5.50600e6i	0.999874i
$498$	3.06979e6i	0.554670i
$499$	−161861.	−0.0290999	−0.0145500	−	0.999894i	$-0.504632\pi$
$499$	−161861.	−0.0290999	−0.0145500	+	0.999894i	$0.504632\pi$
$500$	0	0
$501$	−1.22832e7	−2.18634
$502$	1.86617e6i	0.330515i
$503$	5.40474e6i	0.952477i	0.879316	+	0.476239i	$0.158000\pi$
$503$	5.40474e6i	0.952477i	−0.879316	+	0.476239i	$0.842000\pi$
$504$	−4.41433e6	−0.774085
$505$	0	0
$506$	−2.58256e6	−0.448409
$507$	− 684381.i	− 0.118244i
$508$	− 4.35751e6i	− 0.749169i
$509$	−1.72999e6	−0.295971	−0.147985	−	0.988990i	$-0.547279\pi$
$509$	−1.72999e6	−0.295971	−0.147985	+	0.988990i	$0.547279\pi$
$510$	0	0
$511$	88235.1	0.0149482
$512$	3.84698e6i	0.648553i
$513$	3.41735e6i	0.573318i
$514$	−5.65451e6	−0.944033
$515$	0	0
$516$	−4.76721e6	−0.788207
$517$	5.70510e6i	0.938721i
$518$	505036.i	0.0826985i
$519$	8.69896e6	1.41758
$520$	0	0
$521$	8.15642e6	1.31645	0.658226	−	0.752820i	$-0.271307\pi$
$521$	8.15642e6	1.31645	0.658226	+	0.752820i	$0.271307\pi$
$522$	7.56807e6i	1.21565i
$523$	− 7.57602e6i	− 1.21112i	−0.795800	−	0.605559i	$-0.792949\pi$
$523$	− 7.57602e6i	− 1.21112i	0.795800	−	0.605559i	$-0.207051\pi$
$524$	1.06280e6	0.169093
$525$	0	0
$526$	−4.18245e6	−0.659123
$527$	− 1.20515e6i	− 0.189023i
$528$	3.63820e6i	0.567939i
$529$	1.71248e6	0.266064
$530$	0	0
$531$	−654567.	−0.100744
$532$	3.38318e6i	0.518258i
$533$	− 3.29904e6i	− 0.503002i
$534$	−225280.	−0.0341876
$535$	0	0
$536$	−3.02521e6	−0.454825
$537$	1.94689e7i	2.91344i
$538$	− 498445.i	− 0.0742440i
$539$	4.07656e6	0.604396
$540$	0	0
$541$	1.29055e7	1.89575	0.947876	−	0.318640i	$-0.103226\pi$
$541$	1.29055e7	1.89575	0.947876	+	0.318640i	$0.103226\pi$
$542$	− 4.36108e6i	− 0.637670i
$543$	− 1.11417e7i	− 1.62164i
$544$	−2.60978e6	−0.378099
$545$	0	0
$546$	−957025.	−0.137386
$547$	7.61965e6i	1.08885i	0.838811	+	0.544423i	$0.183252\pi$
$547$	7.61965e6i	1.08885i	−0.838811	+	0.544423i	$0.816748\pi$
$548$	8.47978e6i	1.20624i
$549$	−1.50947e7	−2.13744
$550$	0	0
$551$	1.34068e7	1.88125
$552$	− 8.09727e6i	− 1.13107i
$553$	− 9.22980e6i	− 1.28345i
$554$	−4.23182e6	−0.585805
$555$	0	0
$556$	−1.33215e6	−0.182754
$557$	7.35984e6i	1.00515i	0.864534	+	0.502574i	$0.167614\pi$
$557$	7.35984e6i	1.00515i	−0.864534	+	0.502574i	$0.832386\pi$
$558$	2.50685e6i	0.340833i
$559$	−1.37791e6	−0.186505
$560$	0	0
$561$	−4.53324e6	−0.608137
$562$	2.64416e6i	0.353140i
$563$	− 5.68526e6i	− 0.755927i	−0.925821	−	0.377963i	$-0.876625\pi$
$563$	− 5.68526e6i	− 0.755927i	0.925821	−	0.377963i	$-0.123375\pi$
$564$	−7.73876e6	−1.02441
$565$	0	0
$566$	−7.27744e6	−0.954855
$567$	2.55859e6i	0.334228i
$568$	− 9.98547e6i	− 1.29867i
$569$	1.43108e7	1.85304	0.926519	−	0.376249i	$-0.122786\pi$
$569$	1.43108e7	1.85304	0.926519	+	0.376249i	$0.122786\pi$
$570$	0	0
$571$	−1.95049e6	−0.250354	−0.125177	−	0.992134i	$-0.539950\pi$
$571$	−1.95049e6	−0.250354	−0.125177	+	0.992134i	$0.539950\pi$
$572$	1.77753e6i	0.227158i
$573$	− 372313.i	− 0.0473720i
$574$	−4.61331e6	−0.584430
$575$	0	0
$576$	1.69564e6	0.212950
$577$	987263.i	0.123451i	0.998093	+	0.0617253i	$0.0196603\pi$
$577$	987263.i	0.123451i	−0.998093	+	0.0617253i	$0.980340\pi$
$578$	3.38302e6i	0.421196i
$579$	7.65368e6	0.948798
$580$	0	0
$581$	−3.98417e6	−0.489663
$582$	8.80272e6i	1.07723i
$583$	755697.i	0.0920823i
$584$	−160020.	−0.0194152
$585$	0	0
$586$	−790350.	−0.0950770
$587$	− 1.11813e6i	− 0.133936i	−0.997755	−	0.0669679i	$-0.978668\pi$
$587$	− 1.11813e6i	− 0.133936i	0.997755	−	0.0669679i	$-0.0213325\pi$
$588$	5.52971e6i	0.659567i
$589$	4.44086e6	0.527447
$590$	0	0
$591$	1.52370e7	1.79445
$592$	752746.i	0.0882763i
$593$	− 905001.i	− 0.105685i	−0.998603	−	0.0528424i	$-0.983172\pi$
$593$	− 905001.i	− 0.105685i	0.998603	−	0.0528424i	$-0.0168281\pi$
$594$	2.51077e6	0.291971
$595$	0	0
$596$	6.08874e6	0.702121
$597$	− 1.37606e6i	− 0.158016i
$598$	− 1.01254e6i	− 0.115787i
$599$	5.11859e6	0.582885	0.291443	−	0.956588i	$-0.405865\pi$
$599$	5.11859e6	0.582885	0.291443	+	0.956588i	$0.405865\pi$
$600$	0	0
$601$	8.84991e6	0.999431	0.499716	−	0.866190i	$-0.333438\pi$
$601$	8.84991e6	0.999431	0.499716	+	0.866190i	$0.333438\pi$
$602$	1.92683e6i	0.216697i
$603$	− 6.44403e6i	− 0.721712i
$604$	9.73082e6	1.08532
$605$	0	0
$606$	−7.43957e6	−0.822937
$607$	− 1.37995e7i	− 1.52017i	−0.649824	−	0.760085i	$-0.725157\pi$
$607$	− 1.37995e7i	− 1.52017i	0.649824	−	0.760085i	$-0.274843\pi$
$608$	− 9.61674e6i	− 1.05504i
$609$	−1.70293e7	−1.86061
$610$	0	0
$611$	−2.23679e6	−0.242395
$612$	− 3.54678e6i	− 0.382785i
$613$	1.55631e7i	1.67280i	0.548118	+	0.836401i	$0.315345\pi$
$613$	1.55631e7i	1.67280i	−0.548118	+	0.836401i	$0.684655\pi$
$614$	5.23473e6	0.560368
$615$	0	0
$616$	5.74542e6	0.610057
$617$	− 1.57277e7i	− 1.66323i	−0.555353	−	0.831615i	$-0.687417\pi$
$617$	− 1.57277e7i	− 1.66323i	0.555353	−	0.831615i	$-0.312583\pi$
$618$	− 6.76772e6i	− 0.712806i
$619$	5.68261e6	0.596103	0.298051	−	0.954550i	$-0.403663\pi$
$619$	5.68261e6	0.596103	0.298051	+	0.954550i	$0.403663\pi$
$620$	0	0
$621$	4.59254e6	0.477885
$622$	4.80802e6i	0.498299i
$623$	− 292383.i	− 0.0301809i
$624$	−1.42643e6	−0.146652
$625$	0	0
$626$	−1.66916e6	−0.170240
$627$	− 1.67045e7i	− 1.69693i
$628$	− 1.05436e7i	− 1.06682i
$629$	−937930.	−0.0945244
$630$	0	0
$631$	7.92146e6	0.792012	0.396006	−	0.918248i	$-0.370396\pi$
$631$	7.92146e6	0.792012	0.396006	+	0.918248i	$0.370396\pi$
$632$	1.67388e7i	1.66699i
$633$	− 9.75482e6i	− 0.967631i
$634$	−6.05622e6	−0.598382
$635$	0	0
$636$	−1.02508e6	−0.100488
$637$	1.59829e6i	0.156066i
$638$	− 9.85014e6i	− 0.958055i
$639$	2.12701e7	2.06071
$640$	0	0
$641$	8.63330e6	0.829911	0.414956	−	0.909842i	$-0.363797\pi$
$641$	8.63330e6	0.829911	0.414956	+	0.909842i	$0.363797\pi$
$642$	− 5.97252e6i	− 0.571900i
$643$	5.41901e6i	0.516884i	0.966027	+	0.258442i	$0.0832090\pi$
$643$	5.41901e6i	0.516884i	−0.966027	+	0.258442i	$0.916791\pi$
$644$	4.54662e6	0.431990
$645$	0	0
$646$	1.95670e6	0.184477
$647$	− 2.85189e6i	− 0.267838i	−0.990992	−	0.133919i	$-0.957244\pi$
$647$	− 2.85189e6i	− 0.267838i	0.990992	−	0.133919i	$-0.0427562\pi$
$648$	− 4.64016e6i	− 0.434106i
$649$	851944.	0.0793961
$650$	0	0
$651$	−5.64079e6	−0.521660
$652$	1.37402e6i	0.126582i
$653$	1.31018e7i	1.20239i	0.799101	+	0.601197i	$0.205309\pi$
$653$	1.31018e7i	1.20239i	−0.799101	+	0.601197i	$0.794691\pi$
$654$	1.08935e7	0.995916
$655$	0	0
$656$	−6.87605e6	−0.623849
$657$	− 340859.i	− 0.0308079i
$658$	3.12789e6i	0.281635i
$659$	987681.	0.0885937	0.0442969	−	0.999018i	$-0.485895\pi$
$659$	987681.	0.0885937	0.0442969	+	0.999018i	$0.485895\pi$
$660$	0	0
$661$	2.30367e6	0.205077	0.102538	−	0.994729i	$-0.467304\pi$
$661$	2.30367e6	0.205077	0.102538	+	0.994729i	$0.467304\pi$
$662$	− 1.01500e7i	− 0.900167i
$663$	− 1.77734e6i	− 0.157032i
$664$	7.22553e6	0.635989
$665$	0	0
$666$	1.95099e6	0.170439
$667$	− 1.80172e7i	− 1.56810i
$668$	1.25082e7i	1.08456i
$669$	−2.26056e6	−0.195277
$670$	0	0
$671$	1.96464e7	1.68452
$672$	1.22152e7i	1.04346i
$673$	− 3.89370e6i	− 0.331379i	−0.986178	−	0.165689i	$-0.947015\pi$
$673$	− 3.89370e6i	− 0.331379i	0.986178	−	0.165689i	$-0.0529849\pi$
$674$	−9.55803e6	−0.810436
$675$	0	0
$676$	−696916.	−0.0586562
$677$	− 2.83801e6i	− 0.237981i	−0.992895	−	0.118991i	$-0.962034\pi$
$677$	− 2.83801e6i	− 0.237981i	0.992895	−	0.118991i	$-0.0379658\pi$
$678$	2.40793e6i	0.201173i
$679$	−1.14247e7	−0.950981
$680$	0	0
$681$	6.03595e6	0.498744
$682$	− 3.26276e6i	− 0.268611i
$683$	− 6.98711e6i	− 0.573121i	−0.958062	−	0.286560i	$-0.907488\pi$
$683$	− 6.98711e6i	− 0.573121i	0.958062	−	0.286560i	$-0.0925119\pi$
$684$	1.30695e7	1.06812
$685$	0	0
$686$	6.20695e6	0.503580
$687$	1.02566e7i	0.829105i
$688$	2.87191e6i	0.231313i
$689$	−296286.	−0.0237773
$690$	0	0
$691$	3.58854e6	0.285905	0.142953	−	0.989730i	$-0.454340\pi$
$691$	3.58854e6	0.285905	0.142953	+	0.989730i	$0.454340\pi$
$692$	− 8.85829e6i	− 0.703209i
$693$	1.22384e7i	0.968033i
$694$	1.07931e7	0.850643
$695$	0	0
$696$	3.08838e7	2.41661
$697$	− 8.56764e6i	− 0.668004i
$698$	9.15342e6i	0.711123i
$699$	−1.14955e6	−0.0889888
$700$	0	0
$701$	−1.07723e7	−0.827971	−0.413986	−	0.910283i	$-0.635864\pi$
$701$	−1.07723e7	−0.827971	−0.413986	+	0.910283i	$0.635864\pi$
$702$	984396.i	0.0753923i
$703$	− 3.45617e6i	− 0.263759i
$704$	−2.20694e6	−0.167826
$705$	0	0
$706$	−8.53384e6	−0.644366
$707$	− 9.65556e6i	− 0.726489i
$708$	1.15563e6i	0.0866436i
$709$	−1.68465e7	−1.25862	−0.629309	−	0.777155i	$-0.716662\pi$
$709$	−1.68465e7	−1.25862	−0.629309	+	0.777155i	$0.716662\pi$
$710$	0	0
$711$	−3.56554e7	−2.64516
$712$	530254.i	0.0391998i
$713$	− 5.96803e6i	− 0.439650i
$714$	−2.48540e6	−0.182453
$715$	0	0
$716$	1.98255e7	1.44525
$717$	3.35270e7i	2.43555i
$718$	7.48230e6i	0.541656i
$719$	−7.10379e6	−0.512470	−0.256235	−	0.966615i	$-0.582482\pi$
$719$	−7.10379e6	−0.512470	−0.256235	+	0.966615i	$0.582482\pi$
$720$	0	0
$721$	8.78359e6	0.629265
$722$	384518.i	0.0274520i
$723$	− 1.38211e7i	− 0.983327i
$724$	−1.13458e7	−0.804432
$725$	0	0
$726$	−1.63483e6	−0.115115
$727$	2.50048e7i	1.75464i	0.479908	+	0.877319i	$0.340670\pi$
$727$	2.50048e7i	1.75464i	−0.479908	+	0.877319i	$0.659330\pi$
$728$	2.25260e6i	0.157528i
$729$	2.21877e7	1.54630
$730$	0	0
$731$	−3.57843e6	−0.247685
$732$	2.66496e7i	1.83828i
$733$	− 7.77802e6i	− 0.534699i	−0.963600	−	0.267349i	$-0.913852\pi$
$733$	− 7.77802e6i	− 0.534699i	0.963600	−	0.267349i	$-0.0861477\pi$
$734$	7.02842e6	0.481524
$735$	0	0
$736$	−1.29238e7	−0.879422
$737$	8.38715e6i	0.568782i
$738$	1.78216e7i	1.20450i
$739$	−1.00573e7	−0.677439	−0.338720	−	0.940887i	$-0.609994\pi$
$739$	−1.00573e7	−0.677439	−0.338720	+	0.940887i	$0.609994\pi$
$740$	0	0
$741$	6.54932e6	0.438178
$742$	414320.i	0.0276265i
$743$	− 3.40838e6i	− 0.226504i	−0.993566	−	0.113252i	$-0.963873\pi$
$743$	− 3.40838e6i	− 0.226504i	0.993566	−	0.113252i	$-0.0361268\pi$
$744$	1.02299e7	0.677548
$745$	0	0
$746$	−1.01429e7	−0.667292
$747$	1.53912e7i	1.00918i
$748$	4.61627e6i	0.301673i
$749$	7.75153e6	0.504874
$750$	0	0
$751$	1.57084e7	1.01633	0.508163	−	0.861261i	$-0.330325\pi$
$751$	1.57084e7	1.01633	0.508163	+	0.861261i	$0.330325\pi$
$752$	4.66205e6i	0.300630i
$753$	1.62217e7i	1.04258i
$754$	3.86194e6	0.247387
$755$	0	0
$756$	−4.42023e6	−0.281281
$757$	2.52846e7i	1.60367i	0.597544	+	0.801836i	$0.296143\pi$
$757$	2.52846e7i	1.60367i	−0.597544	+	0.801836i	$0.703857\pi$
$758$	2.94469e6i	0.186152i
$759$	−2.24490e7	−1.41447
$760$	0	0
$761$	1.34808e7	0.843825	0.421913	−	0.906636i	$-0.361359\pi$
$761$	1.34808e7	0.843825	0.421913	+	0.906636i	$0.361359\pi$
$762$	1.17960e7i	0.735949i
$763$	1.41383e7i	0.879195i
$764$	−379132.	−0.0234994
$765$	0	0
$766$	1.65890e6	0.102152
$767$	334021.i	0.0205015i
$768$	− 1.55625e7i	− 0.952087i
$769$	−8.09020e6	−0.493337	−0.246668	−	0.969100i	$-0.579336\pi$
$769$	−8.09020e6	−0.493337	−0.246668	+	0.969100i	$0.579336\pi$
$770$	0	0
$771$	−4.91520e7	−2.97787
$772$	− 7.79387e6i	− 0.470663i
$773$	2.75295e6i	0.165710i	0.996562	+	0.0828551i	$0.0264039\pi$
$773$	2.75295e6i	0.165710i	−0.996562	+	0.0828551i	$0.973596\pi$
$774$	7.44351e6	0.446607
$775$	0	0
$776$	2.07195e7	1.23516
$777$	4.39004e6i	0.260865i
$778$	− 1.37941e7i	− 0.817040i
$779$	3.15708e7	1.86398
$780$	0	0
$781$	−2.76839e7	−1.62405
$782$	− 2.62959e6i	− 0.153770i
$783$	1.75164e7i	1.02103i
$784$	3.33126e6	0.193561
$785$	0	0
$786$	−2.87706e6	−0.166109
$787$	− 2.06536e7i	− 1.18867i	−0.804219	−	0.594333i	$-0.797416\pi$
$787$	− 2.06536e7i	− 1.18867i	0.804219	−	0.594333i	$-0.202584\pi$
$788$	− 1.55161e7i	− 0.890156i
$789$	−3.63560e7	−2.07914
$790$	0	0
$791$	−3.12516e6	−0.177595
$792$	− 2.21950e7i	− 1.25731i
$793$	7.70274e6i	0.434973i
$794$	9.58160e6	0.539370
$795$	0	0
$796$	−1.40126e6	−0.0783855
$797$	− 5.27093e6i	− 0.293928i	−0.989142	−	0.146964i	$-0.953050\pi$
$797$	− 5.27093e6i	− 0.293928i	0.989142	−	0.146964i	$-0.0469502\pi$
$798$	− 9.15843e6i	− 0.509113i
$799$	−5.80897e6	−0.321909
$800$	0	0
$801$	−1.12950e6	−0.0622020
$802$	− 1.41992e7i	− 0.779519i
$803$	443641.i	0.0242797i
$804$	−1.13769e7	−0.620701
$805$	0	0
$806$	1.27923e6	0.0693602
$807$	− 4.33275e6i	− 0.234196i
$808$	1.75110e7i	0.943586i
$809$	2.62432e7	1.40976	0.704882	−	0.709325i	$-0.251000\pi$
$809$	2.62432e7	1.40976	0.704882	+	0.709325i	$0.251000\pi$
$810$	0	0
$811$	1.50454e7	0.803251	0.401626	−	0.915804i	$-0.368445\pi$
$811$	1.50454e7	0.803251	0.401626	+	0.915804i	$0.368445\pi$
$812$	1.73412e7i	0.922975i
$813$	− 3.79088e7i	− 2.01147i
$814$	−2.53929e6	−0.134323
$815$	0	0
$816$	−3.70444e6	−0.194759
$817$	− 1.31861e7i	− 0.691134i
$818$	3.07766e6i	0.160819i
$819$	−4.79829e6	−0.249963
$820$	0	0
$821$	1.82770e6	0.0946338	0.0473169	−	0.998880i	$-0.484933\pi$
$821$	1.82770e6	0.0946338	0.0473169	+	0.998880i	$0.484933\pi$
$822$	− 2.29552e7i	− 1.18495i
$823$	2.45074e7i	1.26124i	0.776093	+	0.630619i	$0.217199\pi$
$823$	2.45074e7i	1.26124i	−0.776093	+	0.630619i	$0.782801\pi$
$824$	−1.59296e7	−0.817309
$825$	0	0
$826$	467088.	0.0238204
$827$	7.03772e6i	0.357823i	0.983865	+	0.178911i	$0.0572576\pi$
$827$	7.03772e6i	0.357823i	−0.983865	+	0.178911i	$0.942742\pi$
$828$	− 1.75640e7i	− 0.890321i
$829$	−3.44967e7	−1.74338	−0.871689	−	0.490059i	$-0.836975\pi$
$829$	−3.44967e7	−1.74338	−0.871689	+	0.490059i	$0.836975\pi$
$830$	0	0
$831$	−3.67852e7	−1.84787
$832$	− 865273.i	− 0.0433356i
$833$	4.15078e6i	0.207261i
$834$	3.60619e6	0.179529
$835$	0	0
$836$	−1.70104e7	−0.841782
$837$	5.80212e6i	0.286268i
$838$	− 4.95476e6i	− 0.243732i
$839$	−2.79128e6	−0.136899	−0.0684493	−	0.997655i	$-0.521805\pi$
$839$	−2.79128e6	−0.136899	−0.0684493	+	0.997655i	$0.521805\pi$
$840$	0	0
$841$	4.82083e7	2.35035
$842$	1.46460e7i	0.711935i
$843$	2.29845e7i	1.11395i
$844$	−9.93349e6	−0.480005
$845$	0	0
$846$	1.20833e7	0.580442
$847$	− 2.12179e6i	− 0.101623i
$848$	617535.i	0.0294898i
$849$	−6.32593e7	−3.01200
$850$	0	0
$851$	−4.64471e6	−0.219854
$852$	− 3.75522e7i	− 1.77230i
$853$	− 2.48047e7i	− 1.16724i	−0.812026	−	0.583621i	$-0.801635\pi$
$853$	− 2.48047e7i	− 1.16724i	0.812026	−	0.583621i	$-0.198365\pi$
$854$	1.07713e7	0.505388
$855$	0	0
$856$	−1.40579e7	−0.655746
$857$	1.83304e7i	0.852550i	0.904594	+	0.426275i	$0.140174\pi$
$857$	1.83304e7i	0.852550i	−0.904594	+	0.426275i	$0.859826\pi$
$858$	− 4.81187e6i	− 0.223149i
$859$	−1.67092e7	−0.772632	−0.386316	−	0.922366i	$-0.626253\pi$
$859$	−1.67092e7	−0.772632	−0.386316	+	0.922366i	$0.626253\pi$
$860$	0	0
$861$	−4.01013e7	−1.84353
$862$	− 1.27806e7i	− 0.585844i
$863$	8.29285e6i	0.379033i	0.981878	+	0.189516i	$0.0606920\pi$
$863$	8.29285e6i	0.379033i	−0.981878	+	0.189516i	$0.939308\pi$
$864$	1.25646e7	0.572615
$865$	0	0
$866$	1.01627e7	0.460484
$867$	2.94070e7i	1.32863i
$868$	5.74411e6i	0.258775i
$869$	4.64069e7	2.08465
$870$	0	0
$871$	−3.28834e6	−0.146870
$872$	− 2.56407e7i	− 1.14193i
$873$	4.41347e7i	1.95995i
$874$	9.68974e6	0.429075
$875$	0	0
$876$	−601784.	−0.0264960
$877$	1.44569e7i	0.634711i	0.948307	+	0.317356i	$0.102795\pi$
$877$	1.44569e7i	0.634711i	−0.948307	+	0.317356i	$0.897205\pi$
$878$	4.60447e6i	0.201578i
$879$	−6.87014e6	−0.299912
$880$	0	0
$881$	−3.42081e7	−1.48487	−0.742437	−	0.669916i	$-0.766330\pi$
$881$	−3.42081e7	−1.48487	−0.742437	+	0.669916i	$0.766330\pi$
$882$	− 8.63407e6i	− 0.373718i
$883$	− 1.41341e7i	− 0.610051i	−0.952344	−	0.305026i	$-0.901335\pi$
$883$	− 1.41341e7i	− 0.610051i	0.952344	−	0.305026i	$-0.0986650\pi$
$884$	−1.80990e6	−0.0778975
$885$	0	0
$886$	−9.90207e6	−0.423781
$887$	− 2.15650e7i	− 0.920325i	−0.887835	−	0.460163i	$-0.847791\pi$
$887$	− 2.15650e7i	− 0.920325i	0.887835	−	0.460163i	$-0.152209\pi$
$888$	− 7.96160e6i	− 0.338819i
$889$	−1.53096e7	−0.649696
$890$	0	0
$891$	−1.28645e7	−0.542872
$892$	2.30196e6i	0.0968693i
$893$	− 2.14054e7i	− 0.898246i
$894$	−1.64825e7	−0.689731
$895$	0	0
$896$	1.51027e7	0.628471
$897$	− 8.80157e6i	− 0.365241i
$898$	− 221785.i	− 0.00917787i
$899$	2.27626e7	0.939340
$900$	0	0
$901$	−769456.	−0.0315771
$902$	− 2.31955e7i	− 0.949264i
$903$	1.67491e7i	0.683551i
$904$	5.66768e6	0.230666
$905$	0	0
$906$	−2.63418e7	−1.06617
$907$	− 1.53494e7i	− 0.619547i	−0.950810	−	0.309773i	$-0.899747\pi$
$907$	− 1.53494e7i	− 0.619547i	0.950810	−	0.309773i	$-0.100253\pi$
$908$	− 6.14650e6i	− 0.247408i
$909$	−3.73002e7	−1.49727
$910$	0	0
$911$	−8.00925e6	−0.319739	−0.159870	−	0.987138i	$-0.551107\pi$
$911$	−8.00925e6	−0.319739	−0.159870	+	0.987138i	$0.551107\pi$
$912$	− 1.36505e7i	− 0.543451i
$913$	− 2.00322e7i	− 0.795337i
$914$	7.93047e6	0.314003
$915$	0	0
$916$	1.04444e7	0.411287
$917$	− 3.73403e6i	− 0.146641i
$918$	2.55648e6i	0.100124i
$919$	1.00787e7	0.393655	0.196828	−	0.980438i	$-0.436936\pi$
$919$	1.00787e7	0.393655	0.196828	+	0.980438i	$0.436936\pi$
$920$	0	0
$921$	4.55031e7	1.76763
$922$	225064.i	0.00871926i
$923$	− 1.08540e7i	− 0.419359i
$924$	2.16067e7	0.832547
$925$	0	0
$926$	6.89748e6	0.264340
$927$	− 3.39317e7i	− 1.29690i
$928$	− 4.92927e7i	− 1.87894i
$929$	1.20808e7	0.459257	0.229629	−	0.973278i	$-0.426249\pi$
$929$	1.20808e7	0.459257	0.229629	+	0.973278i	$0.426249\pi$
$930$	0	0
$931$	−1.52952e7	−0.578336
$932$	1.17061e6i	0.0441440i
$933$	4.17938e7i	1.57184i
$934$	3.82547e6	0.143489
$935$	0	0
$936$	8.70199e6	0.324660
$937$	− 1.85069e7i	− 0.688628i	−0.938854	−	0.344314i	$-0.888111\pi$
$937$	− 1.85069e7i	− 0.688628i	0.938854	−	0.344314i	$-0.111889\pi$
$938$	4.59835e6i	0.170646i
$939$	−1.45092e7	−0.537007
$940$	0	0
$941$	−1.31161e7	−0.482871	−0.241436	−	0.970417i	$-0.577618\pi$
$941$	−1.31161e7	−0.482871	−0.241436	+	0.970417i	$0.577618\pi$
$942$	2.85422e7i	1.04800i
$943$	− 4.24277e7i	− 1.55371i
$944$	696186.	0.0254270
$945$	0	0
$946$	−9.68802e6	−0.351971
$947$	2.46835e7i	0.894399i	0.894434	+	0.447199i	$0.147579\pi$
$947$	2.46835e7i	0.894399i	−0.894434	+	0.447199i	$0.852421\pi$
$948$	6.29494e7i	2.27494i
$949$	−173938.	−0.00626945
$950$	0	0
$951$	−5.26439e7	−1.88754
$952$	5.85003e6i	0.209202i
$953$	3.21978e7i	1.14840i	0.818715	+	0.574201i	$0.194687\pi$
$953$	3.21978e7i	1.14840i	−0.818715	+	0.574201i	$0.805313\pi$
$954$	1.60055e6	0.0569375
$955$	0	0
$956$	3.41411e7	1.20818
$957$	− 8.56226e7i	− 3.02210i
$958$	1.36832e7i	0.481697i
$959$	2.97927e7	1.04608
$960$	0	0
$961$	−2.10893e7	−0.736636
$962$	− 995579.i	− 0.0346847i
$963$	− 2.99448e7i	− 1.04053i
$964$	−1.40743e7	−0.487791
$965$	0	0
$966$	−1.23079e7	−0.424368
$967$	− 2.28108e7i	− 0.784465i	−0.919866	−	0.392232i	$-0.871703\pi$
$967$	− 2.28108e7i	− 0.784465i	0.919866	−	0.392232i	$-0.128297\pi$
$968$	3.84799e6i	0.131991i
$969$	1.70086e7	0.581916
$970$	0	0
$971$	4.02182e7	1.36891	0.684455	−	0.729055i	$-0.260040\pi$
$971$	4.02182e7	1.36891	0.684455	+	0.729055i	$0.260040\pi$
$972$	− 2.99792e7i	− 1.01778i
$973$	4.68035e6i	0.158488i
$974$	2.36346e7	0.798271
$975$	0	0
$976$	1.60545e7	0.539475
$977$	− 3.17409e7i	− 1.06386i	−0.846790	−	0.531928i	$-0.821468\pi$
$977$	− 3.17409e7i	− 1.06386i	0.846790	−	0.531928i	$-0.178532\pi$
$978$	− 3.71953e6i	− 0.124349i
$979$	1.47008e6	0.0490214
$980$	0	0
$981$	5.46173e7	1.81200
$982$	7.03774e6i	0.232892i
$983$	4.52276e6i	0.149286i	0.997210	+	0.0746431i	$0.0237818\pi$
$983$	4.52276e6i	0.149286i	−0.997210	+	0.0746431i	$0.976218\pi$
$984$	7.27262e7	2.39444
$985$	0	0
$986$	1.00295e7	0.328539
$987$	2.71892e7i	0.888391i
$988$	− 6.66928e6i	− 0.217363i
$989$	−1.77207e7	−0.576090
$990$	0	0
$991$	−2.04730e7	−0.662213	−0.331107	−	0.943593i	$-0.607422\pi$
$991$	−2.04730e7	−0.662213	−0.331107	+	0.943593i	$0.607422\pi$
$992$	− 1.63277e7i	− 0.526801i
$993$	− 8.82296e7i	− 2.83949i
$994$	−1.51780e7	−0.487247
$995$	0	0
$996$	2.71729e7	0.867937
$997$	3.49736e7i	1.11430i	0.830412	+	0.557150i	$0.188105\pi$
$997$	3.49736e7i	1.11430i	−0.830412	+	0.557150i	$0.811895\pi$
$998$	− 446192.i	− 0.0141806i
$999$	4.51559e6	0.143153

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.g.274.8		12
5.2	odd	4		65.6.a.d.1.3	✓	6
5.3	odd	4		325.6.a.g.1.4		6
5.4	even	2	inner	325.6.b.g.274.5		12
15.2	even	4		585.6.a.m.1.4		6
20.7	even	4		1040.6.a.q.1.2		6
65.12	odd	4		845.6.a.h.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.d.1.3	✓	6	5.2	odd	4
325.6.a.g.1.4		6	5.3	odd	4
325.6.b.g.274.5		12	5.4	even	2	inner
325.6.b.g.274.8		12	1.1	even	1	trivial
585.6.a.m.1.4		6	15.2	even	4
845.6.a.h.1.4		6	65.12	odd	4
1040.6.a.q.1.2		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+2.75663i q^{2} +23.9621i q^{3} +24.4010 q^{4} -66.0547 q^{6} -85.7300i q^{7} +155.477i q^{8} -331.182 q^{9} +431.046 q^{11} +584.699i q^{12} +169.000i q^{13} +236.326 q^{14} +352.239 q^{16} +438.894i q^{17} -912.946i q^{18} -1617.28 q^{19} +2054.27 q^{21} +1188.23i q^{22} +2173.44i q^{23} -3725.55 q^{24} -465.871 q^{26} -2113.02i q^{27} -2091.90i q^{28} -8289.72 q^{29} -2745.88 q^{31} +5946.25i q^{32} +10328.8i q^{33} -1209.87 q^{34} -8081.16 q^{36} +2137.03i q^{37} -4458.24i q^{38} -4049.59 q^{39} -19520.9 q^{41} +5662.87i q^{42} +8153.29i q^{43} +10517.9 q^{44} -5991.39 q^{46} +13235.5i q^{47} +8440.39i q^{48} +9457.36 q^{49} -10516.8 q^{51} +4123.77i q^{52} +1753.17i q^{53} +5824.83 q^{54} +13329.0 q^{56} -38753.4i q^{57} -22851.7i q^{58} +1976.46 q^{59} +45578.3 q^{61} -7569.39i q^{62} +28392.2i q^{63} -5119.96 q^{64} -28472.6 q^{66} +19457.7i q^{67} +10709.5i q^{68} -52080.3 q^{69} -64224.9 q^{71} -51491.1i q^{72} +1029.22i q^{73} -5891.00 q^{74} -39463.2 q^{76} -36953.6i q^{77} -11163.2i q^{78} +107661. q^{79} -29844.7 q^{81} -53812.1i q^{82} -46473.4i q^{83} +50126.2 q^{84} -22475.6 q^{86} -198639. i q^{87} +67017.6i q^{88} +3410.51 q^{89} +14488.4 q^{91} +53034.2i q^{92} -65797.1i q^{93} -36485.3 q^{94} -142485. q^{96} -133264. i q^{97} +26070.5i q^{98} -142755. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 268 q^{4} + 636 q^{6} - 1036 q^{9} - 340 q^{11} + 2880 q^{14} + 7012 q^{16} - 2436 q^{19} - 792 q^{21} - 25236 q^{24} - 16728 q^{29} + 5724 q^{31} + 42968 q^{34} - 4276 q^{36} - 12844 q^{39} + 4496 q^{41}+ \cdots + 64540 q^{99}+O(q^{100}) \) 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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