LMFDB - Embedded newform 25.6.b.a.24.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [25,6,Mod(24,25)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(25, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("25.24");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 25 = 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	25.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:	$4.00959549532$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 5)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			24.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	25.24
Dual form			25.6.b.a.24.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} +4.00000i q^{3} +28.0000 q^{4} -8.00000 q^{6} +192.000i q^{7} +120.000i q^{8} +227.000 q^{9} +O(q^{10})$	\(q+2.00000i q^{2} +4.00000i q^{3} +28.0000 q^{4} -8.00000 q^{6} +192.000i q^{7} +120.000i q^{8} +227.000 q^{9} -148.000 q^{11} +112.000i q^{12} -286.000i q^{13} -384.000 q^{14} +656.000 q^{16} -1678.00i q^{17} +454.000i q^{18} -1060.00 q^{19} -768.000 q^{21} -296.000i q^{22} -2976.00i q^{23} -480.000 q^{24} +572.000 q^{26} +1880.00i q^{27} +5376.00i q^{28} +3410.00 q^{29} -2448.00 q^{31} +5152.00i q^{32} -592.000i q^{33} +3356.00 q^{34} +6356.00 q^{36} +182.000i q^{37} -2120.00i q^{38} +1144.00 q^{39} -9398.00 q^{41} -1536.00i q^{42} +1244.00i q^{43} -4144.00 q^{44} +5952.00 q^{46} -12088.0i q^{47} +2624.00i q^{48} -20057.0 q^{49} +6712.00 q^{51} -8008.00i q^{52} -23846.0i q^{53} -3760.00 q^{54} -23040.0 q^{56} -4240.00i q^{57} +6820.00i q^{58} +20020.0 q^{59} +32302.0 q^{61} -4896.00i q^{62} +43584.0i q^{63} +10688.0 q^{64} +1184.00 q^{66} +60972.0i q^{67} -46984.0i q^{68} +11904.0 q^{69} -32648.0 q^{71} +27240.0i q^{72} +38774.0i q^{73} -364.000 q^{74} -29680.0 q^{76} -28416.0i q^{77} +2288.00i q^{78} +33360.0 q^{79} +47641.0 q^{81} -18796.0i q^{82} -16716.0i q^{83} -21504.0 q^{84} -2488.00 q^{86} +13640.0i q^{87} -17760.0i q^{88} -101370. q^{89} +54912.0 q^{91} -83328.0i q^{92} -9792.00i q^{93} +24176.0 q^{94} -20608.0 q^{96} -119038. i q^{97} -40114.0i q^{98} -33596.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 56 q^{4} - 16 q^{6} + 454 q^{9}+O(q^{10}) $ 2 * q + 56 * q^4 - 16 * q^6 + 454 * q^9	$ 2 q + 56 q^{4} - 16 q^{6} + 454 q^{9} - 296 q^{11} - 768 q^{14} + 1312 q^{16} - 2120 q^{19} - 1536 q^{21} - 960 q^{24} + 1144 q^{26} + 6820 q^{29} - 4896 q^{31} + 6712 q^{34} + 12712 q^{36} + 2288 q^{39} - 18796 q^{41} - 8288 q^{44} + 11904 q^{46} - 40114 q^{49} + 13424 q^{51} - 7520 q^{54} - 46080 q^{56} + 40040 q^{59} + 64604 q^{61} + 21376 q^{64} + 2368 q^{66} + 23808 q^{69} - 65296 q^{71} - 728 q^{74} - 59360 q^{76} + 66720 q^{79} + 95282 q^{81} - 43008 q^{84} - 4976 q^{86} - 202740 q^{89} + 109824 q^{91} + 48352 q^{94} - 41216 q^{96} - 67192 q^{99}+O(q^{100}) $ 2 * q + 56 * q^4 - 16 * q^6 + 454 * q^9 - 296 * q^11 - 768 * q^14 + 1312 * q^16 - 2120 * q^19 - 1536 * q^21 - 960 * q^24 + 1144 * q^26 + 6820 * q^29 - 4896 * q^31 + 6712 * q^34 + 12712 * q^36 + 2288 * q^39 - 18796 * q^41 - 8288 * q^44 + 11904 * q^46 - 40114 * q^49 + 13424 * q^51 - 7520 * q^54 - 46080 * q^56 + 40040 * q^59 + 64604 * q^61 + 21376 * q^64 + 2368 * q^66 + 23808 * q^69 - 65296 * q^71 - 728 * q^74 - 59360 * q^76 + 66720 * q^79 + 95282 * q^81 - 43008 * q^84 - 4976 * q^86 - 202740 * q^89 + 109824 * q^91 + 48352 * q^94 - 41216 * q^96 - 67192 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/25\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	2.00000i	0.353553i	0.984251	+	0.176777i	$0.0565670\pi$
$2$	2.00000i	0.353553i	−0.984251	+	0.176777i	$0.943433\pi$
$3$	4.00000i	0.256600i	0.991735	+	0.128300i	$0.0409521\pi$
$3$	4.00000i	0.256600i	−0.991735	+	0.128300i	$0.959048\pi$
$4$	28.0000	0.875000
$5$	0	0
$5$	0	0
$6$	−8.00000	−0.0907218
$7$	192.000i	1.48100i	0.672054	+	0.740502i	$0.265412\pi$
$7$	192.000i	1.48100i	−0.672054	+	0.740502i	$0.734588\pi$
$8$	120.000i	0.662913i
$9$	227.000	0.934156
$10$	0	0
$11$	−148.000	−0.368791	−0.184395	−	0.982852i	$-0.559033\pi$
$11$	−148.000	−0.368791	−0.184395	+	0.982852i	$0.559033\pi$
$12$	112.000i	0.224525i
$13$	− 286.000i	− 0.469362i	−0.972072	−	0.234681i	$-0.924595\pi$
$13$	− 286.000i	− 0.469362i	0.972072	−	0.234681i	$-0.0754045\pi$
$14$	−384.000	−0.523614
$15$	0	0
$16$	656.000	0.640625
$17$	− 1678.00i	− 1.40822i	−0.710092	−	0.704109i	$-0.751347\pi$
$17$	− 1678.00i	− 1.40822i	0.710092	−	0.704109i	$-0.248653\pi$
$18$	454.000i	0.330274i
$19$	−1060.00	−0.673631	−0.336815	−	0.941571i	$-0.609350\pi$
$19$	−1060.00	−0.673631	−0.336815	+	0.941571i	$0.609350\pi$
$20$	0	0
$21$	−768.000	−0.380026
$22$	− 296.000i	− 0.130387i
$23$	− 2976.00i	− 1.17304i	−0.809934	−	0.586521i	$-0.800497\pi$
$23$	− 2976.00i	− 1.17304i	0.809934	−	0.586521i	$-0.199503\pi$
$24$	−480.000	−0.170103
$25$	0	0
$26$	572.000	0.165944
$27$	1880.00i	0.496305i
$28$	5376.00i	1.29588i
$29$	3410.00	0.752938	0.376469	−	0.926429i	$-0.377138\pi$
$29$	3410.00	0.752938	0.376469	+	0.926429i	$0.377138\pi$
$30$	0	0
$31$	−2448.00	−0.457517	−0.228758	−	0.973483i	$-0.573467\pi$
$31$	−2448.00	−0.457517	−0.228758	+	0.973483i	$0.573467\pi$
$32$	5152.00i	0.889408i
$33$	− 592.000i	− 0.0946317i
$34$	3356.00	0.497880
$35$	0	0
$36$	6356.00	0.817387
$37$	182.000i	0.0218558i	0.999940	+	0.0109279i	$0.00347853\pi$
$37$	182.000i	0.0218558i	−0.999940	+	0.0109279i	$0.996521\pi$
$38$	− 2120.00i	− 0.238164i
$39$	1144.00	0.120438
$40$	0	0
$41$	−9398.00	−0.873124	−0.436562	−	0.899674i	$-0.643804\pi$
$41$	−9398.00	−0.873124	−0.436562	+	0.899674i	$0.643804\pi$
$42$	− 1536.00i	− 0.134359i
$43$	1244.00i	0.102600i	0.998683	+	0.0513002i	$0.0163365\pi$
$43$	1244.00i	0.102600i	−0.998683	+	0.0513002i	$0.983663\pi$
$44$	−4144.00	−0.322692
$45$	0	0
$46$	5952.00	0.414733
$47$	− 12088.0i	− 0.798196i	−0.916908	−	0.399098i	$-0.869323\pi$
$47$	− 12088.0i	− 0.798196i	0.916908	−	0.399098i	$-0.130677\pi$
$48$	2624.00i	0.164384i
$49$	−20057.0	−1.19337
$50$	0	0
$51$	6712.00	0.361349
$52$	− 8008.00i	− 0.410691i
$53$	− 23846.0i	− 1.16607i	−0.812446	−	0.583037i	$-0.801864\pi$
$53$	− 23846.0i	− 1.16607i	0.812446	−	0.583037i	$-0.198136\pi$
$54$	−3760.00	−0.175470
$55$	0	0
$56$	−23040.0	−0.981776
$57$	− 4240.00i	− 0.172854i
$58$	6820.00i	0.266204i
$59$	20020.0	0.748745	0.374373	−	0.927278i	$-0.377858\pi$
$59$	20020.0	0.748745	0.374373	+	0.927278i	$0.377858\pi$
$60$	0	0
$61$	32302.0	1.11149	0.555744	−	0.831353i	$-0.312433\pi$
$61$	32302.0	1.11149	0.555744	+	0.831353i	$0.312433\pi$
$62$	− 4896.00i	− 0.161757i
$63$	43584.0i	1.38349i
$64$	10688.0	0.326172
$65$	0	0
$66$	1184.00	0.0334574
$67$	60972.0i	1.65937i	0.558231	+	0.829685i	$0.311480\pi$
$67$	60972.0i	1.65937i	−0.558231	+	0.829685i	$0.688520\pi$
$68$	− 46984.0i	− 1.23219i
$69$	11904.0	0.301003
$70$	0	0
$71$	−32648.0	−0.768618	−0.384309	−	0.923204i	$-0.625560\pi$
$71$	−32648.0	−0.768618	−0.384309	+	0.923204i	$0.625560\pi$
$72$	27240.0i	0.619264i
$73$	38774.0i	0.851596i	0.904818	+	0.425798i	$0.140007\pi$
$73$	38774.0i	0.851596i	−0.904818	+	0.425798i	$0.859993\pi$
$74$	−364.000	−0.00772720
$75$	0	0
$76$	−29680.0	−0.589427
$77$	− 28416.0i	− 0.546180i
$78$	2288.00i	0.0425814i
$79$	33360.0	0.601393	0.300696	−	0.953720i	$-0.402781\pi$
$79$	33360.0	0.601393	0.300696	+	0.953720i	$0.402781\pi$
$80$	0	0
$81$	47641.0	0.806805
$82$	− 18796.0i	− 0.308696i
$83$	− 16716.0i	− 0.266340i	−0.991093	−	0.133170i	$-0.957484\pi$
$83$	− 16716.0i	− 0.266340i	0.991093	−	0.133170i	$-0.0425157\pi$
$84$	−21504.0	−0.332522
$85$	0	0
$86$	−2488.00	−0.0362747
$87$	13640.0i	0.193204i
$88$	− 17760.0i	− 0.244476i
$89$	−101370.	−1.35655	−0.678273	−	0.734810i	$-0.737271\pi$
$89$	−101370.	−1.35655	−0.678273	+	0.734810i	$0.737271\pi$
$90$	0	0
$91$	54912.0	0.695126
$92$	− 83328.0i	− 1.02641i
$93$	− 9792.00i	− 0.117399i
$94$	24176.0	0.282205
$95$	0	0
$96$	−20608.0	−0.228222
$97$	− 119038.i	− 1.28457i	−0.766468	−	0.642283i	$-0.777987\pi$
$97$	− 119038.i	− 1.28457i	0.766468	−	0.642283i	$-0.222013\pi$
$98$	− 40114.0i	− 0.421921i
$99$	−33596.0	−0.344508
$100$	0	0
$101$	−89898.0	−0.876893	−0.438446	−	0.898757i	$-0.644471\pi$
$101$	−89898.0	−0.876893	−0.438446	+	0.898757i	$0.644471\pi$
$102$	13424.0i	0.127756i
$103$	19504.0i	0.181147i	0.995890	+	0.0905734i	$0.0288700\pi$
$103$	19504.0i	0.181147i	−0.995890	+	0.0905734i	$0.971130\pi$
$104$	34320.0	0.311146
$105$	0	0
$106$	47692.0	0.412269
$107$	158292.i	1.33659i	0.743895	+	0.668297i	$0.232976\pi$
$107$	158292.i	1.33659i	−0.743895	+	0.668297i	$0.767024\pi$
$108$	52640.0i	0.434267i
$109$	−36830.0	−0.296917	−0.148459	−	0.988919i	$-0.547431\pi$
$109$	−36830.0	−0.296917	−0.148459	+	0.988919i	$0.547431\pi$
$110$	0	0
$111$	−728.000	−0.00560821
$112$	125952.i	0.948768i
$113$	− 11186.0i	− 0.0824098i	−0.999151	−	0.0412049i	$-0.986880\pi$
$113$	− 11186.0i	− 0.0824098i	0.999151	−	0.0412049i	$-0.0131196\pi$
$114$	8480.00	0.0611130
$115$	0	0
$116$	95480.0	0.658821
$117$	− 64922.0i	− 0.438457i
$118$	40040.0i	0.264721i
$119$	322176.	2.08557
$120$	0	0
$121$	−139147.	−0.863993
$122$	64604.0i	0.392970i
$123$	− 37592.0i	− 0.224044i
$124$	−68544.0	−0.400327
$125$	0	0
$126$	−87168.0	−0.489137
$127$	70552.0i	0.388150i	0.980987	+	0.194075i	$0.0621706\pi$
$127$	70552.0i	0.388150i	−0.980987	+	0.194075i	$0.937829\pi$
$128$	186240.i	1.00473i
$129$	−4976.00	−0.0263273
$130$	0	0
$131$	76452.0	0.389234	0.194617	−	0.980879i	$-0.437654\pi$
$131$	76452.0	0.389234	0.194617	+	0.980879i	$0.437654\pi$
$132$	− 16576.0i	− 0.0828028i
$133$	− 203520.i	− 0.997650i
$134$	−121944.	−0.586676
$135$	0	0
$136$	201360.	0.933525
$137$	− 144918.i	− 0.659661i	−0.944040	−	0.329831i	$-0.893008\pi$
$137$	− 144918.i	− 0.659661i	0.944040	−	0.329831i	$-0.106992\pi$
$138$	23808.0i	0.106420i
$139$	−112220.	−0.492644	−0.246322	−	0.969188i	$-0.579222\pi$
$139$	−112220.	−0.492644	−0.246322	+	0.969188i	$0.579222\pi$
$140$	0	0
$141$	48352.0	0.204817
$142$	− 65296.0i	− 0.271748i
$143$	42328.0i	0.173096i
$144$	148912.	0.598444
$145$	0	0
$146$	−77548.0	−0.301085
$147$	− 80228.0i	− 0.306219i
$148$	5096.00i	0.0191238i
$149$	−403750.	−1.48986	−0.744932	−	0.667140i	$-0.767518\pi$
$149$	−403750.	−1.48986	−0.744932	+	0.667140i	$0.767518\pi$
$150$	0	0
$151$	−446648.	−1.59413	−0.797064	−	0.603895i	$-0.793615\pi$
$151$	−446648.	−1.59413	−0.797064	+	0.603895i	$0.793615\pi$
$152$	− 127200.i	− 0.446558i
$153$	− 380906.i	− 1.31550i
$154$	56832.0	0.193104
$155$	0	0
$156$	32032.0	0.105383
$157$	− 262258.i	− 0.849141i	−0.905395	−	0.424570i	$-0.860425\pi$
$157$	− 262258.i	− 0.849141i	0.905395	−	0.424570i	$-0.139575\pi$
$158$	66720.0i	0.212625i
$159$	95384.0	0.299215
$160$	0	0
$161$	571392.	1.73728
$162$	95282.0i	0.285248i
$163$	154564.i	0.455658i	0.973701	+	0.227829i	$0.0731628\pi$
$163$	154564.i	0.455658i	−0.973701	+	0.227829i	$0.926837\pi$
$164$	−263144.	−0.763983
$165$	0	0
$166$	33432.0	0.0941656
$167$	396672.i	1.10063i	0.834958	+	0.550314i	$0.185492\pi$
$167$	396672.i	1.10063i	−0.834958	+	0.550314i	$0.814508\pi$
$168$	− 92160.0i	− 0.251924i
$169$	289497.	0.779700
$170$	0	0
$171$	−240620.	−0.629276
$172$	34832.0i	0.0897754i
$173$	573474.i	1.45680i	0.685155	+	0.728398i	$0.259735\pi$
$173$	573474.i	1.45680i	−0.685155	+	0.728398i	$0.740265\pi$
$174$	−27280.0	−0.0683079
$175$	0	0
$176$	−97088.0	−0.236257
$177$	80080.0i	0.192128i
$178$	− 202740.i	− 0.479611i
$179$	594460.	1.38672	0.693362	−	0.720589i	$-0.256129\pi$
$179$	594460.	1.38672	0.693362	+	0.720589i	$0.256129\pi$
$180$	0	0
$181$	−107098.	−0.242988	−0.121494	−	0.992592i	$-0.538769\pi$
$181$	−107098.	−0.242988	−0.121494	+	0.992592i	$0.538769\pi$
$182$	109824.i	0.245764i
$183$	129208.i	0.285208i
$184$	357120.	0.777624
$185$	0	0
$186$	19584.0	0.0415068
$187$	248344.i	0.519337i
$188$	− 338464.i	− 0.698422i
$189$	−360960.	−0.735029
$190$	0	0
$191$	469552.	0.931323	0.465661	−	0.884963i	$-0.345816\pi$
$191$	469552.	0.931323	0.465661	+	0.884963i	$0.345816\pi$
$192$	42752.0i	0.0836957i
$193$	− 52706.0i	− 0.101851i	−0.998702	−	0.0509257i	$-0.983783\pi$
$193$	− 52706.0i	− 0.101851i	0.998702	−	0.0509257i	$-0.0162172\pi$
$194$	238076.	0.454163
$195$	0	0
$196$	−561596.	−1.04420
$197$	455862.i	0.836889i	0.908242	+	0.418444i	$0.137425\pi$
$197$	455862.i	0.836889i	−0.908242	+	0.418444i	$0.862575\pi$
$198$	− 67192.0i	− 0.121802i
$199$	−865000.	−1.54840	−0.774200	−	0.632940i	$-0.781848\pi$
$199$	−865000.	−1.54840	−0.774200	+	0.632940i	$0.781848\pi$
$200$	0	0
$201$	−243888.	−0.425795
$202$	− 179796.i	− 0.310028i
$203$	654720.i	1.11510i
$204$	187936.	0.316180
$205$	0	0
$206$	−39008.0	−0.0640451
$207$	− 675552.i	− 1.09580i
$208$	− 187616.i	− 0.300685i
$209$	156880.	0.248429
$210$	0	0
$211$	1.10565e6	1.70967	0.854835	−	0.518900i	$-0.173658\pi$
$211$	1.10565e6	1.70967	0.854835	+	0.518900i	$0.173658\pi$
$212$	− 667688.i	− 1.02031i
$213$	− 130592.i	− 0.197228i
$214$	−316584.	−0.472557
$215$	0	0
$216$	−225600.	−0.329007
$217$	− 470016.i	− 0.677584i
$218$	− 73660.0i	− 0.104976i
$219$	−155096.	−0.218520
$220$	0	0
$221$	−479908.	−0.660963
$222$	− 1456.00i	− 0.00198280i
$223$	− 1.12158e6i	− 1.51031i	−0.655545	−	0.755156i	$-0.727561\pi$
$223$	− 1.12158e6i	− 1.51031i	0.655545	−	0.755156i	$-0.272439\pi$
$224$	−989184.	−1.31722
$225$	0	0
$226$	22372.0	0.0291363
$227$	− 23348.0i	− 0.0300736i	−0.999887	−	0.0150368i	$-0.995213\pi$
$227$	− 23348.0i	− 0.0300736i	0.999887	−	0.0150368i	$-0.00478654\pi$
$228$	− 118720.i	− 0.151247i
$229$	596010.	0.751043	0.375522	−	0.926814i	$-0.377464\pi$
$229$	596010.	0.751043	0.375522	+	0.926814i	$0.377464\pi$
$230$	0	0
$231$	113664.	0.140150
$232$	409200.i	0.499132i
$233$	485334.i	0.585667i	0.956163	+	0.292834i	$0.0945982\pi$
$233$	485334.i	0.585667i	−0.956163	+	0.292834i	$0.905402\pi$
$234$	129844.	0.155018
$235$	0	0
$236$	560560.	0.655152
$237$	133440.i	0.154317i
$238$	644352.i	0.737362i
$239$	48880.0	0.0553524	0.0276762	−	0.999617i	$-0.491189\pi$
$239$	48880.0	0.0553524	0.0276762	+	0.999617i	$0.491189\pi$
$240$	0	0
$241$	−110798.	−0.122882	−0.0614411	−	0.998111i	$-0.519570\pi$
$241$	−110798.	−0.122882	−0.0614411	+	0.998111i	$0.519570\pi$
$242$	− 278294.i	− 0.305468i
$243$	647404.i	0.703331i
$244$	904456.	0.972552
$245$	0	0
$246$	75184.0	0.0792114
$247$	303160.i	0.316176i
$248$	− 293760.i	− 0.303294i
$249$	66864.0	0.0683430
$250$	0	0
$251$	−1.64375e6	−1.64684	−0.823419	−	0.567434i	$-0.807936\pi$
$251$	−1.64375e6	−1.64684	−0.823419	+	0.567434i	$0.807936\pi$
$252$	1.22035e6i	1.21055i
$253$	440448.i	0.432607i
$254$	−141104.	−0.137232
$255$	0	0
$256$	−30464.0	−0.0290527
$257$	1.30624e6i	1.23365i	0.787102	+	0.616823i	$0.211581\pi$
$257$	1.30624e6i	1.23365i	−0.787102	+	0.616823i	$0.788419\pi$
$258$	− 9952.00i	− 0.00930810i
$259$	−34944.0	−0.0323685
$260$	0	0
$261$	774070.	0.703362
$262$	152904.i	0.137615i
$263$	− 2.12834e6i	− 1.89736i	−0.316231	−	0.948682i	$-0.602417\pi$
$263$	− 2.12834e6i	− 1.89736i	0.316231	−	0.948682i	$-0.397583\pi$
$264$	71040.0	0.0627326
$265$	0	0
$266$	407040.	0.352722
$267$	− 405480.i	− 0.348090i
$268$	1.70722e6i	1.45195i
$269$	1.44109e6	1.21426	0.607128	−	0.794604i	$-0.292321\pi$
$269$	1.44109e6	1.21426	0.607128	+	0.794604i	$0.292321\pi$
$270$	0	0
$271$	−93248.0	−0.0771288	−0.0385644	−	0.999256i	$-0.512278\pi$
$271$	−93248.0	−0.0771288	−0.0385644	+	0.999256i	$0.512278\pi$
$272$	− 1.10077e6i	− 0.902139i
$273$	219648.i	0.178370i
$274$	289836.	0.233225
$275$	0	0
$276$	333312.	0.263377
$277$	− 110298.i	− 0.0863711i	−0.999067	−	0.0431855i	$-0.986249\pi$
$277$	− 110298.i	− 0.0863711i	0.999067	−	0.0431855i	$-0.0137507\pi$
$278$	− 224440.i	− 0.174176i
$279$	−555696.	−0.427392
$280$	0	0
$281$	−192198.	−0.145205	−0.0726027	−	0.997361i	$-0.523131\pi$
$281$	−192198.	−0.145205	−0.0726027	+	0.997361i	$0.523131\pi$
$282$	96704.0i	0.0724139i
$283$	331884.i	0.246332i	0.992386	+	0.123166i	$0.0393047\pi$
$283$	331884.i	0.246332i	−0.992386	+	0.123166i	$0.960695\pi$
$284$	−914144.	−0.672541
$285$	0	0
$286$	−84656.0	−0.0611988
$287$	− 1.80442e6i	− 1.29310i
$288$	1.16950e6i	0.830846i
$289$	−1.39583e6	−0.983076
$290$	0	0
$291$	476152.	0.329620
$292$	1.08567e6i	0.745146i
$293$	− 2.19481e6i	− 1.49358i	−0.665063	−	0.746788i	$-0.731595\pi$
$293$	− 2.19481e6i	− 1.49358i	0.665063	−	0.746788i	$-0.268405\pi$
$294$	160456.	0.108265
$295$	0	0
$296$	−21840.0	−0.0144885
$297$	− 278240.i	− 0.183033i
$298$	− 807500.i	− 0.526747i
$299$	−851136.	−0.550581
$300$	0	0
$301$	−238848.	−0.151952
$302$	− 893296.i	− 0.563609i
$303$	− 359592.i	− 0.225011i
$304$	−695360.	−0.431545
$305$	0	0
$306$	761812.	0.465098
$307$	− 2.37751e6i	− 1.43971i	−0.694123	−	0.719857i	$-0.744207\pi$
$307$	− 2.37751e6i	− 1.43971i	0.694123	−	0.719857i	$-0.255793\pi$
$308$	− 795648.i	− 0.477908i
$309$	−78016.0	−0.0464823
$310$	0	0
$311$	−2.37305e6	−1.39125	−0.695626	−	0.718405i	$-0.744873\pi$
$311$	−2.37305e6	−1.39125	−0.695626	+	0.718405i	$0.744873\pi$
$312$	137280.i	0.0798400i
$313$	1.42941e6i	0.824702i	0.911025	+	0.412351i	$0.135292\pi$
$313$	1.42941e6i	0.824702i	−0.911025	+	0.412351i	$0.864708\pi$
$314$	524516.	0.300217
$315$	0	0
$316$	934080.	0.526219
$317$	2.12462e6i	1.18750i	0.804650	+	0.593750i	$0.202353\pi$
$317$	2.12462e6i	1.18750i	−0.804650	+	0.593750i	$0.797647\pi$
$318$	190768.i	0.105788i
$319$	−504680.	−0.277677
$320$	0	0
$321$	−633168.	−0.342970
$322$	1.14278e6i	0.614221i
$323$	1.77868e6i	0.948618i
$324$	1.33395e6	0.705954
$325$	0	0
$326$	−309128.	−0.161100
$327$	− 147320.i	− 0.0761890i
$328$	− 1.12776e6i	− 0.578805i
$329$	2.32090e6	1.18213
$330$	0	0
$331$	3.09985e6	1.55515	0.777573	−	0.628793i	$-0.216451\pi$
$331$	3.09985e6	1.55515	0.777573	+	0.628793i	$0.216451\pi$
$332$	− 468048.i	− 0.233048i
$333$	41314.0i	0.0204168i
$334$	−793344.	−0.389131
$335$	0	0
$336$	−503808.	−0.243454
$337$	2.40008e6i	1.15120i	0.817731	+	0.575601i	$0.195232\pi$
$337$	2.40008e6i	1.15120i	−0.817731	+	0.575601i	$0.804768\pi$
$338$	578994.i	0.275665i
$339$	44744.0	0.0211464
$340$	0	0
$341$	362304.	0.168728
$342$	− 481240.i	− 0.222483i
$343$	− 624000.i	− 0.286384i
$344$	−149280.	−0.0680151
$345$	0	0
$346$	−1.14695e6	−0.515055
$347$	1.77741e6i	0.792436i	0.918156	+	0.396218i	$0.129678\pi$
$347$	1.77741e6i	0.792436i	−0.918156	+	0.396218i	$0.870322\pi$
$348$	381920.i	0.169054i
$349$	2.14805e6	0.944019	0.472010	−	0.881593i	$-0.343529\pi$
$349$	2.14805e6	0.944019	0.472010	+	0.881593i	$0.343529\pi$
$350$	0	0
$351$	537680.	0.232946
$352$	− 762496.i	− 0.328005i
$353$	661854.i	0.282700i	0.989960	+	0.141350i	$0.0451443\pi$
$353$	661854.i	0.282700i	−0.989960	+	0.141350i	$0.954856\pi$
$354$	−160160.	−0.0679275
$355$	0	0
$356$	−2.83836e6	−1.18698
$357$	1.28870e6i	0.535159i
$358$	1.18892e6i	0.490281i
$359$	259320.	0.106194	0.0530970	−	0.998589i	$-0.483091\pi$
$359$	259320.	0.106194	0.0530970	+	0.998589i	$0.483091\pi$
$360$	0	0
$361$	−1.35250e6	−0.546222
$362$	− 214196.i	− 0.0859093i
$363$	− 556588.i	− 0.221701i
$364$	1.53754e6	0.608236
$365$	0	0
$366$	−258416.	−0.100836
$367$	− 1.49993e6i	− 0.581307i	−0.956828	−	0.290653i	$-0.906127\pi$
$367$	− 1.49993e6i	− 0.581307i	0.956828	−	0.290653i	$-0.0938726\pi$
$368$	− 1.95226e6i	− 0.751480i
$369$	−2.13335e6	−0.815634
$370$	0	0
$371$	4.57843e6	1.72696
$372$	− 274176.i	− 0.102724i
$373$	2.23807e6i	0.832918i	0.909154	+	0.416459i	$0.136729\pi$
$373$	2.23807e6i	0.832918i	−0.909154	+	0.416459i	$0.863271\pi$
$374$	−496688.	−0.183614
$375$	0	0
$376$	1.45056e6	0.529135
$377$	− 975260.i	− 0.353400i
$378$	− 721920.i	− 0.259872i
$379$	−3.15934e6	−1.12979	−0.564896	−	0.825162i	$-0.691084\pi$
$379$	−3.15934e6	−1.12979	−0.564896	+	0.825162i	$0.691084\pi$
$380$	0	0
$381$	−282208.	−0.0995994
$382$	939104.i	0.329272i
$383$	− 342216.i	− 0.119207i	−0.998222	−	0.0596037i	$-0.981016\pi$
$383$	− 342216.i	− 0.119207i	0.998222	−	0.0596037i	$-0.0189837\pi$
$384$	−744960.	−0.257813
$385$	0	0
$386$	105412.	0.0360099
$387$	282388.i	0.0958449i
$388$	− 3.33306e6i	− 1.12399i
$389$	−88470.0	−0.0296430	−0.0148215	−	0.999890i	$-0.504718\pi$
$389$	−88470.0	−0.0296430	−0.0148215	+	0.999890i	$0.504718\pi$
$390$	0	0
$391$	−4.99373e6	−1.65190
$392$	− 2.40684e6i	− 0.791101i
$393$	305808.i	0.0998775i
$394$	−911724.	−0.295885
$395$	0	0
$396$	−940688.	−0.301445
$397$	− 5.45674e6i	− 1.73763i	−0.495138	−	0.868814i	$-0.664883\pi$
$397$	− 5.45674e6i	− 1.73763i	0.495138	−	0.868814i	$-0.335117\pi$
$398$	− 1.73000e6i	− 0.547442i
$399$	814080.	0.255997
$400$	0	0
$401$	4.04680e6	1.25676	0.628378	−	0.777908i	$-0.283719\pi$
$401$	4.04680e6	1.25676	0.628378	+	0.777908i	$0.283719\pi$
$402$	− 487776.i	− 0.150541i
$403$	700128.i	0.214741i
$404$	−2.51714e6	−0.767281
$405$	0	0
$406$	−1.30944e6	−0.394249
$407$	− 26936.0i	− 0.00806022i
$408$	805440.i	0.239543i
$409$	2.71207e6	0.801664	0.400832	−	0.916151i	$-0.368721\pi$
$409$	2.71207e6	0.801664	0.400832	+	0.916151i	$0.368721\pi$
$410$	0	0
$411$	579672.	0.169269
$412$	546112.i	0.158503i
$413$	3.84384e6i	1.10889i
$414$	1.35110e6	0.387425
$415$	0	0
$416$	1.47347e6	0.417454
$417$	− 448880.i	− 0.126413i
$418$	313760.i	0.0878328i
$419$	−3.71746e6	−1.03445	−0.517227	−	0.855848i	$-0.673036\pi$
$419$	−3.71746e6	−1.03445	−0.517227	+	0.855848i	$0.673036\pi$
$420$	0	0
$421$	3.55250e6	0.976853	0.488426	−	0.872605i	$-0.337571\pi$
$421$	3.55250e6	0.976853	0.488426	+	0.872605i	$0.337571\pi$
$422$	2.21130e6i	0.604460i
$423$	− 2.74398e6i	− 0.745640i
$424$	2.86152e6	0.773005
$425$	0	0
$426$	261184.	0.0697305
$427$	6.20198e6i	1.64612i
$428$	4.43218e6i	1.16952i
$429$	−169312.	−0.0444165
$430$	0	0
$431$	−4.06205e6	−1.05330	−0.526650	−	0.850082i	$-0.676552\pi$
$431$	−4.06205e6	−1.05330	−0.526650	+	0.850082i	$0.676552\pi$
$432$	1.23328e6i	0.317945i
$433$	− 7.26287e6i	− 1.86161i	−0.365518	−	0.930804i	$-0.619108\pi$
$433$	− 7.26287e6i	− 1.86161i	0.365518	−	0.930804i	$-0.380892\pi$
$434$	940032.	0.239562
$435$	0	0
$436$	−1.03124e6	−0.259803
$437$	3.15456e6i	0.790197i
$438$	− 310192.i	− 0.0772583i
$439$	5.41028e6	1.33986	0.669928	−	0.742426i	$-0.266325\pi$
$439$	5.41028e6	1.33986	0.669928	+	0.742426i	$0.266325\pi$
$440$	0	0
$441$	−4.55294e6	−1.11480
$442$	− 959816.i	− 0.233686i
$443$	6.51524e6i	1.57733i	0.614826	+	0.788663i	$0.289226\pi$
$443$	6.51524e6i	1.57733i	−0.614826	+	0.788663i	$0.710774\pi$
$444$	−20384.0	−0.00490718
$445$	0	0
$446$	2.24315e6	0.533976
$447$	− 1.61500e6i	− 0.382299i
$448$	2.05210e6i	0.483062i
$449$	509950.	0.119375	0.0596873	−	0.998217i	$-0.480990\pi$
$449$	509950.	0.119375	0.0596873	+	0.998217i	$0.480990\pi$
$450$	0	0
$451$	1.39090e6	0.322000
$452$	− 313208.i	− 0.0721085i
$453$	− 1.78659e6i	− 0.409053i
$454$	46696.0	0.0106326
$455$	0	0
$456$	508800.	0.114587
$457$	1.22084e6i	0.273444i	0.990609	+	0.136722i	$0.0436568\pi$
$457$	1.22084e6i	0.273444i	−0.990609	+	0.136722i	$0.956343\pi$
$458$	1.19202e6i	0.265534i
$459$	3.15464e6	0.698905
$460$	0	0
$461$	−4.07210e6	−0.892413	−0.446207	−	0.894930i	$-0.647225\pi$
$461$	−4.07210e6	−0.892413	−0.446207	+	0.894930i	$0.647225\pi$
$462$	227328.i	0.0495505i
$463$	− 2.02294e6i	− 0.438561i	−0.975662	−	0.219280i	$-0.929629\pi$
$463$	− 2.02294e6i	− 0.438561i	0.975662	−	0.219280i	$-0.0703709\pi$
$464$	2.23696e6	0.482351
$465$	0	0
$466$	−970668.	−0.207065
$467$	3.25097e6i	0.689797i	0.938640	+	0.344898i	$0.112087\pi$
$467$	3.25097e6i	0.689797i	−0.938640	+	0.344898i	$0.887913\pi$
$468$	− 1.81782e6i	− 0.383650i
$469$	−1.17066e7	−2.45753
$470$	0	0
$471$	1.04903e6	0.217890
$472$	2.40240e6i	0.496353i
$473$	− 184112.i	− 0.0378381i
$474$	−266880.	−0.0545595
$475$	0	0
$476$	9.02093e6	1.82488
$477$	− 5.41304e6i	− 1.08929i
$478$	97760.0i	0.0195700i
$479$	3.27936e6	0.653056	0.326528	−	0.945188i	$-0.394121\pi$
$479$	3.27936e6	0.653056	0.326528	+	0.945188i	$0.394121\pi$
$480$	0	0
$481$	52052.0	0.0102583
$482$	− 221596.i	− 0.0434455i
$483$	2.28557e6i	0.445786i
$484$	−3.89612e6	−0.755994
$485$	0	0
$486$	−1.29481e6	−0.248665
$487$	− 8.53197e6i	− 1.63015i	−0.579357	−	0.815074i	$-0.696696\pi$
$487$	− 8.53197e6i	− 1.63015i	0.579357	−	0.815074i	$-0.303304\pi$
$488$	3.87624e6i	0.736819i
$489$	−618256.	−0.116922
$490$	0	0
$491$	1.51265e6	0.283162	0.141581	−	0.989927i	$-0.454781\pi$
$491$	1.51265e6	0.283162	0.141581	+	0.989927i	$0.454781\pi$
$492$	− 1.05258e6i	− 0.196038i
$493$	− 5.72198e6i	− 1.06030i
$494$	−606320.	−0.111785
$495$	0	0
$496$	−1.60589e6	−0.293097
$497$	− 6.26842e6i	− 1.13833i
$498$	133728.i	0.0241629i
$499$	6.49190e6	1.16713	0.583567	−	0.812065i	$-0.301657\pi$
$499$	6.49190e6	1.16713	0.583567	+	0.812065i	$0.301657\pi$
$500$	0	0
$501$	−1.58669e6	−0.282421
$502$	− 3.28750e6i	− 0.582245i
$503$	− 8.61770e6i	− 1.51870i	−0.650684	−	0.759349i	$-0.725518\pi$
$503$	− 8.61770e6i	− 1.51870i	0.650684	−	0.759349i	$-0.274482\pi$
$504$	−5.23008e6	−0.917132
$505$	0	0
$506$	−880896.	−0.152950
$507$	1.15799e6i	0.200071i
$508$	1.97546e6i	0.339632i
$509$	−2.67323e6	−0.457343	−0.228671	−	0.973504i	$-0.573438\pi$
$509$	−2.67323e6	−0.457343	−0.228671	+	0.973504i	$0.573438\pi$
$510$	0	0
$511$	−7.44461e6	−1.26122
$512$	5.89875e6i	0.994455i
$513$	− 1.99280e6i	− 0.334326i
$514$	−2.61248e6	−0.436160
$515$	0	0
$516$	−139328.	−0.0230364
$517$	1.78902e6i	0.294367i
$518$	− 69888.0i	− 0.0114440i
$519$	−2.29390e6	−0.373814
$520$	0	0
$521$	6.18500e6	0.998264	0.499132	−	0.866526i	$-0.333652\pi$
$521$	6.18500e6	0.998264	0.499132	+	0.866526i	$0.333652\pi$
$522$	1.54814e6i	0.248676i
$523$	6.89452e6i	1.10217i	0.834448	+	0.551087i	$0.185787\pi$
$523$	6.89452e6i	1.10217i	−0.834448	+	0.551087i	$0.814213\pi$
$524$	2.14066e6	0.340580
$525$	0	0
$526$	4.25667e6	0.670820
$527$	4.10774e6i	0.644283i
$528$	− 388352.i	− 0.0606235i
$529$	−2.42023e6	−0.376026
$530$	0	0
$531$	4.54454e6	0.699445
$532$	− 5.69856e6i	− 0.872943i
$533$	2.68783e6i	0.409811i
$534$	810960.	0.123068
$535$	0	0
$536$	−7.31664e6	−1.10002
$537$	2.37784e6i	0.355834i
$538$	2.88218e6i	0.429304i
$539$	2.96844e6	0.440104
$540$	0	0
$541$	155502.	0.0228425	0.0114212	−	0.999935i	$-0.496364\pi$
$541$	155502.	0.0228425	0.0114212	+	0.999935i	$0.496364\pi$
$542$	− 186496.i	− 0.0272691i
$543$	− 428392.i	− 0.0623508i
$544$	8.64506e6	1.25248
$545$	0	0
$546$	−439296.	−0.0630631
$547$	1.26544e7i	1.80831i	0.427201	+	0.904157i	$0.359500\pi$
$547$	1.26544e7i	1.80831i	−0.427201	+	0.904157i	$0.640500\pi$
$548$	− 4.05770e6i	− 0.577204i
$549$	7.33255e6	1.03830
$550$	0	0
$551$	−3.61460e6	−0.507202
$552$	1.42848e6i	0.199538i
$553$	6.40512e6i	0.890665i
$554$	220596.	0.0305368
$555$	0	0
$556$	−3.14216e6	−0.431064
$557$	− 7.07786e6i	− 0.966638i	−0.875444	−	0.483319i	$-0.839431\pi$
$557$	− 7.07786e6i	− 0.966638i	0.875444	−	0.483319i	$-0.160569\pi$
$558$	− 1.11139e6i	− 0.151106i
$559$	355784.	0.0481567
$560$	0	0
$561$	−993376.	−0.133262
$562$	− 384396.i	− 0.0513379i
$563$	− 846636.i	− 0.112571i	−0.998415	−	0.0562854i	$-0.982074\pi$
$563$	− 846636.i	− 0.112571i	0.998415	−	0.0562854i	$-0.0179257\pi$
$564$	1.35386e6	0.179215
$565$	0	0
$566$	−663768.	−0.0870914
$567$	9.14707e6i	1.19488i
$568$	− 3.91776e6i	− 0.509527i
$569$	−4.96041e6	−0.642299	−0.321149	−	0.947029i	$-0.604069\pi$
$569$	−4.96041e6	−0.642299	−0.321149	+	0.947029i	$0.604069\pi$
$570$	0	0
$571$	8.96505e6	1.15070	0.575351	−	0.817907i	$-0.304866\pi$
$571$	8.96505e6	1.15070	0.575351	+	0.817907i	$0.304866\pi$
$572$	1.18518e6i	0.151459i
$573$	1.87821e6i	0.238978i
$574$	3.60883e6	0.457180
$575$	0	0
$576$	2.42618e6	0.304696
$577$	− 2.86080e6i	− 0.357724i	−0.983874	−	0.178862i	$-0.942758\pi$
$577$	− 2.86080e6i	− 0.357724i	0.983874	−	0.178862i	$-0.0572415\pi$
$578$	− 2.79165e6i	− 0.347570i
$579$	210824.	0.0261351
$580$	0	0
$581$	3.20947e6	0.394451
$582$	952304.i	0.116538i
$583$	3.52921e6i	0.430037i
$584$	−4.65288e6	−0.564534
$585$	0	0
$586$	4.38961e6	0.528059
$587$	− 6.74027e6i	− 0.807387i	−0.914894	−	0.403694i	$-0.867726\pi$
$587$	− 6.74027e6i	− 0.807387i	0.914894	−	0.403694i	$-0.132274\pi$
$588$	− 2.24638e6i	− 0.267942i
$589$	2.59488e6	0.308197
$590$	0	0
$591$	−1.82345e6	−0.214746
$592$	119392.i	0.0140014i
$593$	1.78609e6i	0.208578i	0.994547	+	0.104289i	$0.0332566\pi$
$593$	1.78609e6i	0.208578i	−0.994547	+	0.104289i	$0.966743\pi$
$594$	556480.	0.0647118
$595$	0	0
$596$	−1.13050e7	−1.30363
$597$	− 3.46000e6i	− 0.397320i
$598$	− 1.70227e6i	− 0.194660i
$599$	−4.94620e6	−0.563254	−0.281627	−	0.959524i	$-0.590874\pi$
$599$	−4.94620e6	−0.563254	−0.281627	+	0.959524i	$0.590874\pi$
$600$	0	0
$601$	−4.58100e6	−0.517337	−0.258669	−	0.965966i	$-0.583284\pi$
$601$	−4.58100e6	−0.517337	−0.258669	+	0.965966i	$0.583284\pi$
$602$	− 477696.i	− 0.0537230i
$603$	1.38406e7i	1.55011i
$604$	−1.25061e7	−1.39486
$605$	0	0
$606$	719184.	0.0795533
$607$	7.07999e6i	0.779940i	0.920828	+	0.389970i	$0.127515\pi$
$607$	7.07999e6i	0.779940i	−0.920828	+	0.389970i	$0.872485\pi$
$608$	− 5.46112e6i	− 0.599132i
$609$	−2.61888e6	−0.286136
$610$	0	0
$611$	−3.45717e6	−0.374643
$612$	− 1.06654e7i	− 1.15106i
$613$	− 5.09609e6i	− 0.547754i	−0.961765	−	0.273877i	$-0.911694\pi$
$613$	− 5.09609e6i	− 0.547754i	0.961765	−	0.273877i	$-0.0883061\pi$
$614$	4.75502e6	0.509016
$615$	0	0
$616$	3.40992e6	0.362070
$617$	− 1.30003e7i	− 1.37480i	−0.726279	−	0.687400i	$-0.758752\pi$
$617$	− 1.30003e7i	− 1.37480i	0.726279	−	0.687400i	$-0.241248\pi$
$618$	− 156032.i	− 0.0164340i
$619$	−4.84406e6	−0.508139	−0.254070	−	0.967186i	$-0.581769\pi$
$619$	−4.84406e6	−0.508139	−0.254070	+	0.967186i	$0.581769\pi$
$620$	0	0
$621$	5.59488e6	0.582186
$622$	− 4.74610e6i	− 0.491882i
$623$	− 1.94630e7i	− 2.00905i
$624$	750464.	0.0771558
$625$	0	0
$626$	−2.85883e6	−0.291576
$627$	627520.i	0.0637468i
$628$	− 7.34322e6i	− 0.742998i
$629$	305396.	0.0307777
$630$	0	0
$631$	6.22775e6	0.622670	0.311335	−	0.950300i	$-0.399224\pi$
$631$	6.22775e6	0.622670	0.311335	+	0.950300i	$0.399224\pi$
$632$	4.00320e6i	0.398671i
$633$	4.42261e6i	0.438702i
$634$	−4.24924e6	−0.419845
$635$	0	0
$636$	2.67075e6	0.261813
$637$	5.73630e6i	0.560123i
$638$	− 1.00936e6i	− 0.0981735i
$639$	−7.41110e6	−0.718010
$640$	0	0
$641$	1.53280e6	0.147347	0.0736734	−	0.997282i	$-0.476528\pi$
$641$	1.53280e6	0.147347	0.0736734	+	0.997282i	$0.476528\pi$
$642$	− 1.26634e6i	− 0.121258i
$643$	1.74382e7i	1.66332i	0.555287	+	0.831659i	$0.312609\pi$
$643$	1.74382e7i	1.66332i	−0.555287	+	0.831659i	$0.687391\pi$
$644$	1.59990e7	1.52012
$645$	0	0
$646$	−3.55736e6	−0.335387
$647$	− 4.25469e6i	− 0.399583i	−0.979838	−	0.199792i	$-0.935974\pi$
$647$	− 4.25469e6i	− 0.399583i	0.979838	−	0.199792i	$-0.0640265\pi$
$648$	5.71692e6i	0.534841i
$649$	−2.96296e6	−0.276130
$650$	0	0
$651$	1.88006e6	0.173868
$652$	4.32779e6i	0.398701i
$653$	− 3.01085e6i	− 0.276316i	−0.990410	−	0.138158i	$-0.955882\pi$
$653$	− 3.01085e6i	− 0.276316i	0.990410	−	0.138158i	$-0.0441181\pi$
$654$	294640.	0.0269369
$655$	0	0
$656$	−6.16509e6	−0.559345
$657$	8.80170e6i	0.795524i
$658$	4.64179e6i	0.417947i
$659$	8.11462e6	0.727871	0.363936	−	0.931424i	$-0.381433\pi$
$659$	8.11462e6	0.727871	0.363936	+	0.931424i	$0.381433\pi$
$660$	0	0
$661$	2.47370e6	0.220213	0.110107	−	0.993920i	$-0.464881\pi$
$661$	2.47370e6	0.220213	0.110107	+	0.993920i	$0.464881\pi$
$662$	6.19970e6i	0.549827i
$663$	− 1.91963e6i	− 0.169603i
$664$	2.00592e6	0.176560
$665$	0	0
$666$	−82628.0	−0.00721841
$667$	− 1.01482e7i	− 0.883228i
$668$	1.11068e7i	0.963049i
$669$	4.48630e6	0.387546
$670$	0	0
$671$	−4.78070e6	−0.409907
$672$	− 3.95674e6i	− 0.337998i
$673$	− 5.77063e6i	− 0.491117i	−0.969382	−	0.245559i	$-0.921029\pi$
$673$	− 5.77063e6i	− 0.491117i	0.969382	−	0.245559i	$-0.0789714\pi$
$674$	−4.80016e6	−0.407011
$675$	0	0
$676$	8.10592e6	0.682237
$677$	1.67197e7i	1.40203i	0.713147	+	0.701014i	$0.247269\pi$
$677$	1.67197e7i	1.40203i	−0.713147	+	0.701014i	$0.752731\pi$
$678$	89488.0i	0.00747637i
$679$	2.28553e7	1.90245
$680$	0	0
$681$	93392.0	0.00771688
$682$	724608.i	0.0596544i
$683$	− 7.14532e6i	− 0.586097i	−0.956098	−	0.293049i	$-0.905330\pi$
$683$	− 7.14532e6i	− 0.586097i	0.956098	−	0.293049i	$-0.0946698\pi$
$684$	−6.73736e6	−0.550617
$685$	0	0
$686$	1.24800e6	0.101252
$687$	2.38404e6i	0.192718i
$688$	816064.i	0.0657284i
$689$	−6.81996e6	−0.547310
$690$	0	0
$691$	−8.78395e6	−0.699833	−0.349917	−	0.936781i	$-0.613790\pi$
$691$	−8.78395e6	−0.699833	−0.349917	+	0.936781i	$0.613790\pi$
$692$	1.60573e7i	1.27470i
$693$	− 6.45043e6i	− 0.510218i
$694$	−3.55482e6	−0.280169
$695$	0	0
$696$	−1.63680e6	−0.128077
$697$	1.57698e7i	1.22955i
$698$	4.29610e6i	0.333761i
$699$	−1.94134e6	−0.150282
$700$	0	0
$701$	−1.60141e7	−1.23086	−0.615428	−	0.788193i	$-0.711017\pi$
$701$	−1.60141e7	−1.23086	−0.615428	+	0.788193i	$0.711017\pi$
$702$	1.07536e6i	0.0823590i
$703$	− 192920.i	− 0.0147228i
$704$	−1.58182e6	−0.120289
$705$	0	0
$706$	−1.32371e6	−0.0999495
$707$	− 1.72604e7i	− 1.29868i
$708$	2.24224e6i	0.168112i
$709$	1.91354e7	1.42962	0.714811	−	0.699318i	$-0.246513\pi$
$709$	1.91354e7	1.42962	0.714811	+	0.699318i	$0.246513\pi$
$710$	0	0
$711$	7.57272e6	0.561795
$712$	− 1.21644e7i	− 0.899271i
$713$	7.28525e6i	0.536686i
$714$	−2.57741e6	−0.189207
$715$	0	0
$716$	1.66449e7	1.21338
$717$	195520.i	0.0142034i
$718$	518640.i	0.0375452i
$719$	−1.02934e7	−0.742566	−0.371283	−	0.928520i	$-0.621082\pi$
$719$	−1.02934e7	−0.742566	−0.371283	+	0.928520i	$0.621082\pi$
$720$	0	0
$721$	−3.74477e6	−0.268279
$722$	− 2.70500e6i	− 0.193119i
$723$	− 443192.i	− 0.0315316i
$724$	−2.99874e6	−0.212615
$725$	0	0
$726$	1.11318e6	0.0783831
$727$	− 1.93264e7i	− 1.35618i	−0.734981	−	0.678088i	$-0.762809\pi$
$727$	− 1.93264e7i	− 1.35618i	0.734981	−	0.678088i	$-0.237191\pi$
$728$	6.58944e6i	0.460808i
$729$	8.98715e6	0.626330
$730$	0	0
$731$	2.08743e6	0.144484
$732$	3.61782e6i	0.249557i
$733$	− 5.26197e6i	− 0.361733i	−0.983508	−	0.180866i	$-0.942110\pi$
$733$	− 5.26197e6i	− 0.361733i	0.983508	−	0.180866i	$-0.0578902\pi$
$734$	2.99986e6	0.205523
$735$	0	0
$736$	1.53324e7	1.04331
$737$	− 9.02386e6i	− 0.611961i
$738$	− 4.26669e6i	− 0.288370i
$739$	−2.82944e7	−1.90585	−0.952927	−	0.303199i	$-0.901945\pi$
$739$	−2.82944e7	−1.90585	−0.952927	+	0.303199i	$0.901945\pi$
$740$	0	0
$741$	−1.21264e6	−0.0811309
$742$	9.15686e6i	0.610572i
$743$	− 2.09863e7i	− 1.39464i	−0.716759	−	0.697321i	$-0.754375\pi$
$743$	− 2.09863e7i	− 1.39464i	0.716759	−	0.697321i	$-0.245625\pi$
$744$	1.17504e6	0.0778252
$745$	0	0
$746$	−4.47615e6	−0.294481
$747$	− 3.79453e6i	− 0.248804i
$748$	6.95363e6i	0.454420i
$749$	−3.03921e7	−1.97950
$750$	0	0
$751$	−1.89668e7	−1.22714	−0.613572	−	0.789639i	$-0.710268\pi$
$751$	−1.89668e7	−1.22714	−0.613572	+	0.789639i	$0.710268\pi$
$752$	− 7.92973e6i	− 0.511345i
$753$	− 6.57499e6i	− 0.422579i
$754$	1.95052e6	0.124946
$755$	0	0
$756$	−1.01069e7	−0.643151
$757$	− 1.08257e7i	− 0.686617i	−0.939223	−	0.343309i	$-0.888452\pi$
$757$	− 1.08257e7i	− 0.686617i	0.939223	−	0.343309i	$-0.111548\pi$
$758$	− 6.31868e6i	− 0.399442i
$759$	−1.76179e6	−0.111007
$760$	0	0
$761$	1.90534e7	1.19264	0.596322	−	0.802745i	$-0.296628\pi$
$761$	1.90534e7	1.19264	0.596322	+	0.802745i	$0.296628\pi$
$762$	− 564416.i	− 0.0352137i
$763$	− 7.07136e6i	− 0.439736i
$764$	1.31475e7	0.814908
$765$	0	0
$766$	684432.	0.0421462
$767$	− 5.72572e6i	− 0.351432i
$768$	− 121856.i	− 0.00745494i
$769$	1.57826e7	0.962415	0.481208	−	0.876607i	$-0.340198\pi$
$769$	1.57826e7	0.962415	0.481208	+	0.876607i	$0.340198\pi$
$770$	0	0
$771$	−5.22497e6	−0.316554
$772$	− 1.47577e6i	− 0.0891199i
$773$	2.44049e7i	1.46902i	0.678598	+	0.734510i	$0.262588\pi$
$773$	2.44049e7i	1.46902i	−0.678598	+	0.734510i	$0.737412\pi$
$774$	−564776.	−0.0338863
$775$	0	0
$776$	1.42846e7	0.851555
$777$	− 139776.i	− 0.00830577i
$778$	− 176940.i	− 0.0104804i
$779$	9.96188e6	0.588163
$780$	0	0
$781$	4.83190e6	0.283459
$782$	− 9.98746e6i	− 0.584034i
$783$	6.41080e6i	0.373687i
$784$	−1.31574e7	−0.764504
$785$	0	0
$786$	−611616.	−0.0353120
$787$	3.37607e7i	1.94301i	0.237019	+	0.971505i	$0.423830\pi$
$787$	3.37607e7i	1.94301i	−0.237019	+	0.971505i	$0.576170\pi$
$788$	1.27641e7i	0.732278i
$789$	8.51334e6	0.486864
$790$	0	0
$791$	2.14771e6	0.122049
$792$	− 4.03152e6i	− 0.228379i
$793$	− 9.23837e6i	− 0.521690i
$794$	1.09135e7	0.614344
$795$	0	0
$796$	−2.42200e7	−1.35485
$797$	2.19885e7i	1.22617i	0.790019	+	0.613083i	$0.210071\pi$
$797$	2.19885e7i	1.22617i	−0.790019	+	0.613083i	$0.789929\pi$
$798$	1.62816e6i	0.0905086i
$799$	−2.02837e7	−1.12403
$800$	0	0
$801$	−2.30110e7	−1.26723
$802$	8.09360e6i	0.444330i
$803$	− 5.73855e6i	− 0.314061i
$804$	−6.82886e6	−0.372570
$805$	0	0
$806$	−1.40026e6	−0.0759224
$807$	5.76436e6i	0.311578i
$808$	− 1.07878e7i	− 0.581303i
$809$	2.93597e7	1.57717	0.788587	−	0.614923i	$-0.210813\pi$
$809$	2.93597e7	1.57717	0.788587	+	0.614923i	$0.210813\pi$
$810$	0	0
$811$	3.17703e7	1.69617	0.848083	−	0.529863i	$-0.177757\pi$
$811$	3.17703e7	1.69617	0.848083	+	0.529863i	$0.177757\pi$
$812$	1.83322e7i	0.975716i
$813$	− 372992.i	− 0.0197912i
$814$	53872.0	0.00284972
$815$	0	0
$816$	4.40307e6	0.231489
$817$	− 1.31864e6i	− 0.0691148i
$818$	5.42414e6i	0.283431i
$819$	1.24650e7	0.649357
$820$	0	0
$821$	−2.71430e6	−0.140540	−0.0702699	−	0.997528i	$-0.522386\pi$
$821$	−2.71430e6	−0.140540	−0.0702699	+	0.997528i	$0.522386\pi$
$822$	1.15934e6i	0.0598457i
$823$	1.25866e7i	0.647753i	0.946099	+	0.323877i	$0.104986\pi$
$823$	1.25866e7i	0.647753i	−0.946099	+	0.323877i	$0.895014\pi$
$824$	−2.34048e6	−0.120084
$825$	0	0
$826$	−7.68768e6	−0.392053
$827$	− 8.72355e6i	− 0.443537i	−0.975099	−	0.221768i	$-0.928817\pi$
$827$	− 8.72355e6i	− 0.443537i	0.975099	−	0.221768i	$-0.0711828\pi$
$828$	− 1.89155e7i	− 0.958829i
$829$	1.06178e7	0.536597	0.268299	−	0.963336i	$-0.413539\pi$
$829$	1.06178e7	0.536597	0.268299	+	0.963336i	$0.413539\pi$
$830$	0	0
$831$	441192.	0.0221628
$832$	− 3.05677e6i	− 0.153093i
$833$	3.36556e7i	1.68053i
$834$	897760.	0.0446936
$835$	0	0
$836$	4.39264e6	0.217375
$837$	− 4.60224e6i	− 0.227068i
$838$	− 7.43492e6i	− 0.365735i
$839$	−1.67765e7	−0.822805	−0.411403	−	0.911454i	$-0.634961\pi$
$839$	−1.67765e7	−0.822805	−0.411403	+	0.911454i	$0.634961\pi$
$840$	0	0
$841$	−8.88305e6	−0.433084
$842$	7.10500e6i	0.345370i
$843$	− 768792.i	− 0.0372597i
$844$	3.09583e7	1.49596
$845$	0	0
$846$	5.48795e6	0.263624
$847$	− 2.67162e7i	− 1.27958i
$848$	− 1.56430e7i	− 0.747016i
$849$	−1.32754e6	−0.0632087
$850$	0	0
$851$	541632.	0.0256378
$852$	− 3.65658e6i	− 0.172574i
$853$	2.20186e7i	1.03613i	0.855340	+	0.518067i	$0.173348\pi$
$853$	2.20186e7i	1.03613i	−0.855340	+	0.518067i	$0.826652\pi$
$854$	−1.24040e7	−0.581991
$855$	0	0
$856$	−1.89950e7	−0.886045
$857$	3.16676e7i	1.47287i	0.676510	+	0.736434i	$0.263492\pi$
$857$	3.16676e7i	1.47287i	−0.676510	+	0.736434i	$0.736508\pi$
$858$	− 338624.i	− 0.0157036i
$859$	−1.58064e7	−0.730886	−0.365443	−	0.930834i	$-0.619082\pi$
$859$	−1.58064e7	−0.730886	−0.365443	+	0.930834i	$0.619082\pi$
$860$	0	0
$861$	7.21766e6	0.331809
$862$	− 8.12410e6i	− 0.372398i
$863$	1.44287e7i	0.659476i	0.944072	+	0.329738i	$0.106960\pi$
$863$	1.44287e7i	0.659476i	−0.944072	+	0.329738i	$0.893040\pi$
$864$	−9.68576e6	−0.441417
$865$	0	0
$866$	1.45257e7	0.658178
$867$	− 5.58331e6i	− 0.252257i
$868$	− 1.31604e7i	− 0.592886i
$869$	−4.93728e6	−0.221788
$870$	0	0
$871$	1.74380e7	0.778845
$872$	− 4.41960e6i	− 0.196830i
$873$	− 2.70216e7i	− 1.19999i
$874$	−6.30912e6	−0.279377
$875$	0	0
$876$	−4.34269e6	−0.191205
$877$	247902.i	0.0108838i	0.999985	+	0.00544191i	$0.00173222\pi$
$877$	247902.i	0.0108838i	−0.999985	+	0.00544191i	$0.998268\pi$
$878$	1.08206e7i	0.473711i
$879$	8.77922e6	0.383252
$880$	0	0
$881$	4.10268e7	1.78085	0.890426	−	0.455128i	$-0.150406\pi$
$881$	4.10268e7	1.78085	0.890426	+	0.455128i	$0.150406\pi$
$882$	− 9.10588e6i	− 0.394140i
$883$	− 4.18015e7i	− 1.80422i	−0.431503	−	0.902112i	$-0.642016\pi$
$883$	− 4.18015e7i	− 1.80422i	0.431503	−	0.902112i	$-0.357984\pi$
$884$	−1.34374e7	−0.578343
$885$	0	0
$886$	−1.30305e7	−0.557669
$887$	− 2.10476e7i	− 0.898241i	−0.893471	−	0.449120i	$-0.851737\pi$
$887$	− 2.10476e7i	− 0.898241i	0.893471	−	0.449120i	$-0.148263\pi$
$888$	− 87360.0i	− 0.00371775i
$889$	−1.35460e7	−0.574852
$890$	0	0
$891$	−7.05087e6	−0.297542
$892$	− 3.14041e7i	− 1.32152i
$893$	1.28133e7i	0.537690i
$894$	3.23000e6	0.135163
$895$	0	0
$896$	−3.57581e7	−1.48800
$897$	− 3.40454e6i	− 0.141279i
$898$	1.01990e6i	0.0422053i
$899$	−8.34768e6	−0.344482
$900$	0	0
$901$	−4.00136e7	−1.64208
$902$	2.78181e6i	0.113844i
$903$	− 955392.i	− 0.0389908i
$904$	1.34232e6	0.0546305
$905$	0	0
$906$	3.57318e6	0.144622
$907$	7.48309e6i	0.302039i	0.988531	+	0.151019i	$0.0482556\pi$
$907$	7.48309e6i	0.302039i	−0.988531	+	0.151019i	$0.951744\pi$
$908$	− 653744.i	− 0.0263144i
$909$	−2.04068e7	−0.819155
$910$	0	0
$911$	−6.63165e6	−0.264744	−0.132372	−	0.991200i	$-0.542259\pi$
$911$	−6.63165e6	−0.264744	−0.132372	+	0.991200i	$0.542259\pi$
$912$	− 2.78144e6i	− 0.110734i
$913$	2.47397e6i	0.0982239i
$914$	−2.44168e6	−0.0966772
$915$	0	0
$916$	1.66883e7	0.657163
$917$	1.46788e7i	0.576457i
$918$	6.30928e6i	0.247100i
$919$	1.68976e7	0.659990	0.329995	−	0.943983i	$-0.392953\pi$
$919$	1.68976e7	0.659990	0.329995	+	0.943983i	$0.392953\pi$
$920$	0	0
$921$	9.51003e6	0.369431
$922$	− 8.14420e6i	− 0.315516i
$923$	9.33733e6i	0.360760i
$924$	3.18259e6	0.122631
$925$	0	0
$926$	4.04587e6	0.155055
$927$	4.42741e6i	0.169219i
$928$	1.75683e7i	0.669669i
$929$	1.28653e7	0.489081	0.244541	−	0.969639i	$-0.421363\pi$
$929$	1.28653e7	0.489081	0.244541	+	0.969639i	$0.421363\pi$
$930$	0	0
$931$	2.12604e7	0.803892
$932$	1.35894e7i	0.512459i
$933$	− 9.49219e6i	− 0.356995i
$934$	−6.50194e6	−0.243880
$935$	0	0
$936$	7.79064e6	0.290659
$937$	1.06887e7i	0.397718i	0.980028	+	0.198859i	$0.0637236\pi$
$937$	1.06887e7i	0.397718i	−0.980028	+	0.198859i	$0.936276\pi$
$938$	− 2.34132e7i	− 0.868870i
$939$	−5.71766e6	−0.211619
$940$	0	0
$941$	2.82455e7	1.03986	0.519930	−	0.854209i	$-0.325958\pi$
$941$	2.82455e7	1.03986	0.519930	+	0.854209i	$0.325958\pi$
$942$	2.09806e6i	0.0770356i
$943$	2.79684e7i	1.02421i
$944$	1.31331e7	0.479665
$945$	0	0
$946$	368224.	0.0133778
$947$	− 1.70892e7i	− 0.619222i	−0.950863	−	0.309611i	$-0.899801\pi$
$947$	− 1.70892e7i	− 0.619222i	0.950863	−	0.309611i	$-0.100199\pi$
$948$	3.73632e6i	0.135028i
$949$	1.10894e7	0.399706
$950$	0	0
$951$	−8.49849e6	−0.304713
$952$	3.86611e7i	1.38255i
$953$	− 2.22259e7i	− 0.792735i	−0.918092	−	0.396367i	$-0.870271\pi$
$953$	− 2.22259e7i	− 0.792735i	0.918092	−	0.396367i	$-0.129729\pi$
$954$	1.08261e7	0.385124
$955$	0	0
$956$	1.36864e6	0.0484333
$957$	− 2.01872e6i	− 0.0712519i
$958$	6.55872e6i	0.230890i
$959$	2.78243e7	0.976961
$960$	0	0
$961$	−2.26364e7	−0.790678
$962$	104104.i	0.00362685i
$963$	3.59323e7i	1.24859i
$964$	−3.10234e6	−0.107522
$965$	0	0
$966$	−4.57114e6	−0.157609
$967$	2.41551e7i	0.830696i	0.909663	+	0.415348i	$0.136340\pi$
$967$	2.41551e7i	0.830696i	−0.909663	+	0.415348i	$0.863660\pi$
$968$	− 1.66976e7i	− 0.572752i
$969$	−7.11472e6	−0.243416
$970$	0	0
$971$	−5.48313e7	−1.86630	−0.933149	−	0.359491i	$-0.882950\pi$
$971$	−5.48313e7	−1.86630	−0.933149	+	0.359491i	$0.882950\pi$
$972$	1.81273e7i	0.615415i
$973$	− 2.15462e7i	− 0.729608i
$974$	1.70639e7	0.576344
$975$	0	0
$976$	2.11901e7	0.712047
$977$	− 1.56612e7i	− 0.524915i	−0.964944	−	0.262457i	$-0.915467\pi$
$977$	− 1.56612e7i	− 0.524915i	0.964944	−	0.262457i	$-0.0845329\pi$
$978$	− 1.23651e6i	− 0.0413382i
$979$	1.50028e7	0.500281
$980$	0	0
$981$	−8.36041e6	−0.277367
$982$	3.02530e6i	0.100113i
$983$	1.63420e7i	0.539412i	0.962943	+	0.269706i	$0.0869266\pi$
$983$	1.63420e7i	0.539412i	−0.962943	+	0.269706i	$0.913073\pi$
$984$	4.51104e6	0.148521
$985$	0	0
$986$	1.14440e7	0.374873
$987$	9.28358e6i	0.303335i
$988$	8.48848e6i	0.276654i
$989$	3.70214e6	0.120355
$990$	0	0
$991$	1.37576e7	0.444997	0.222498	−	0.974933i	$-0.428579\pi$
$991$	1.37576e7	0.444997	0.222498	+	0.974933i	$0.428579\pi$
$992$	− 1.26121e7i	− 0.406919i
$993$	1.23994e7i	0.399050i
$994$	1.25368e7	0.402459
$995$	0	0
$996$	1.87219e6	0.0598001
$997$	− 1.29097e7i	− 0.411320i	−0.978624	−	0.205660i	$-0.934066\pi$
$997$	− 1.29097e7i	− 0.411320i	0.978624	−	0.205660i	$-0.0659341\pi$
$998$	1.29838e7i	0.412644i
$999$	−342160.	−0.0108471

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.b.a.24.2		2
3.2	odd	2		225.6.b.e.199.1		2
4.3	odd	2		400.6.c.j.49.1		2
5.2	odd	4		25.6.a.a.1.1		1
5.3	odd	4		5.6.a.a.1.1	✓	1
5.4	even	2	inner	25.6.b.a.24.1		2
15.2	even	4		225.6.a.f.1.1		1
15.8	even	4		45.6.a.b.1.1		1
15.14	odd	2		225.6.b.e.199.2		2
20.3	even	4		80.6.a.e.1.1		1
20.7	even	4		400.6.a.g.1.1		1
20.19	odd	2		400.6.c.j.49.2		2
35.13	even	4		245.6.a.b.1.1		1
40.3	even	4		320.6.a.g.1.1		1
40.13	odd	4		320.6.a.j.1.1		1
55.43	even	4		605.6.a.a.1.1		1
60.23	odd	4		720.6.a.a.1.1		1
65.38	odd	4		845.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.6.a.a.1.1	✓	1	5.3	odd	4
25.6.a.a.1.1		1	5.2	odd	4
25.6.b.a.24.1		2	5.4	even	2	inner
25.6.b.a.24.2		2	1.1	even	1	trivial
45.6.a.b.1.1		1	15.8	even	4
80.6.a.e.1.1		1	20.3	even	4
225.6.a.f.1.1		1	15.2	even	4
225.6.b.e.199.1		2	3.2	odd	2
225.6.b.e.199.2		2	15.14	odd	2
245.6.a.b.1.1		1	35.13	even	4
320.6.a.g.1.1		1	40.3	even	4
320.6.a.j.1.1		1	40.13	odd	4
400.6.a.g.1.1		1	20.7	even	4
400.6.c.j.49.1		2	4.3	odd	2
400.6.c.j.49.2		2	20.19	odd	2
605.6.a.a.1.1		1	55.43	even	4
720.6.a.a.1.1		1	60.23	odd	4
845.6.a.b.1.1		1	65.38	odd	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:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	25.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +4.00000i q^{3} +28.0000 q^{4} -8.00000 q^{6} +192.000i q^{7} +120.000i q^{8} +227.000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{2} +4.00000i q^{3} +28.0000 q^{4} -8.00000 q^{6} +192.000i q^{7} +120.000i q^{8} +227.000 q^{9} -148.000 q^{11} +112.000i q^{12} -286.000i q^{13} -384.000 q^{14} +656.000 q^{16} -1678.00i q^{17} +454.000i q^{18} -1060.00 q^{19} -768.000 q^{21} -296.000i q^{22} -2976.00i q^{23} -480.000 q^{24} +572.000 q^{26} +1880.00i q^{27} +5376.00i q^{28} +3410.00 q^{29} -2448.00 q^{31} +5152.00i q^{32} -592.000i q^{33} +3356.00 q^{34} +6356.00 q^{36} +182.000i q^{37} -2120.00i q^{38} +1144.00 q^{39} -9398.00 q^{41} -1536.00i q^{42} +1244.00i q^{43} -4144.00 q^{44} +5952.00 q^{46} -12088.0i q^{47} +2624.00i q^{48} -20057.0 q^{49} +6712.00 q^{51} -8008.00i q^{52} -23846.0i q^{53} -3760.00 q^{54} -23040.0 q^{56} -4240.00i q^{57} +6820.00i q^{58} +20020.0 q^{59} +32302.0 q^{61} -4896.00i q^{62} +43584.0i q^{63} +10688.0 q^{64} +1184.00 q^{66} +60972.0i q^{67} -46984.0i q^{68} +11904.0 q^{69} -32648.0 q^{71} +27240.0i q^{72} +38774.0i q^{73} -364.000 q^{74} -29680.0 q^{76} -28416.0i q^{77} +2288.00i q^{78} +33360.0 q^{79} +47641.0 q^{81} -18796.0i q^{82} -16716.0i q^{83} -21504.0 q^{84} -2488.00 q^{86} +13640.0i q^{87} -17760.0i q^{88} -101370. q^{89} +54912.0 q^{91} -83328.0i q^{92} -9792.00i q^{93} +24176.0 q^{94} -20608.0 q^{96} -119038. i q^{97} -40114.0i q^{98} -33596.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 56 q^{4} - 16 q^{6} + 454 q^{9}+O(q^{10}) \) 2 * q + 56 * q^4 - 16 * q^6 + 454 * q^9	\( 2 q + 56 q^{4} - 16 q^{6} + 454 q^{9} - 296 q^{11} - 768 q^{14} + 1312 q^{16} - 2120 q^{19} - 1536 q^{21} - 960 q^{24} + 1144 q^{26} + 6820 q^{29} - 4896 q^{31} + 6712 q^{34} + 12712 q^{36} + 2288 q^{39} - 18796 q^{41} - 8288 q^{44} + 11904 q^{46} - 40114 q^{49} + 13424 q^{51} - 7520 q^{54} - 46080 q^{56} + 40040 q^{59} + 64604 q^{61} + 21376 q^{64} + 2368 q^{66} + 23808 q^{69} - 65296 q^{71} - 728 q^{74} - 59360 q^{76} + 66720 q^{79} + 95282 q^{81} - 43008 q^{84} - 4976 q^{86} - 202740 q^{89} + 109824 q^{91} + 48352 q^{94} - 41216 q^{96} - 67192 q^{99}+O(q^{100}) \) 2 * q + 56 * q^4 - 16 * q^6 + 454 * q^9 - 296 * q^11 - 768 * q^14 + 1312 * q^16 - 2120 * q^19 - 1536 * q^21 - 960 * q^24 + 1144 * q^26 + 6820 * q^29 - 4896 * q^31 + 6712 * q^34 + 12712 * q^36 + 2288 * q^39 - 18796 * q^41 - 8288 * q^44 + 11904 * q^46 - 40114 * q^49 + 13424 * q^51 - 7520 * q^54 - 46080 * q^56 + 40040 * q^59 + 64604 * q^61 + 21376 * q^64 + 2368 * q^66 + 23808 * q^69 - 65296 * q^71 - 728 * q^74 - 59360 * q^76 + 66720 * q^79 + 95282 * q^81 - 43008 * q^84 - 4976 * q^86 - 202740 * q^89 + 109824 * q^91 + 48352 * q^94 - 41216 * q^96 - 67192 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more