LMFDB - Embedded newform 350.6.c.h.99.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,6,Mod(99,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("350.99");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	350.c (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:	$56.1343369345$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	350.99
Dual form			350.6.c.h.99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} -23.0000i q^{3} -16.0000 q^{4} +92.0000 q^{6} -49.0000i q^{7} -64.0000i q^{8} -286.000 q^{9} +O(q^{10})$	\(q+4.00000i q^{2} -23.0000i q^{3} -16.0000 q^{4} +92.0000 q^{6} -49.0000i q^{7} -64.0000i q^{8} -286.000 q^{9} +555.000 q^{11} +368.000i q^{12} -241.000i q^{13} +196.000 q^{14} +256.000 q^{16} +1491.00i q^{17} -1144.00i q^{18} +2038.00 q^{19} -1127.00 q^{21} +2220.00i q^{22} -1230.00i q^{23} -1472.00 q^{24} +964.000 q^{26} +989.000i q^{27} +784.000i q^{28} +5001.00 q^{29} +5696.00 q^{31} +1024.00i q^{32} -12765.0i q^{33} -5964.00 q^{34} +4576.00 q^{36} +5602.00i q^{37} +8152.00i q^{38} -5543.00 q^{39} -2424.00 q^{41} -4508.00i q^{42} +602.000i q^{43} -8880.00 q^{44} +4920.00 q^{46} +23163.0i q^{47} -5888.00i q^{48} -2401.00 q^{49} +34293.0 q^{51} +3856.00i q^{52} -25296.0i q^{53} -3956.00 q^{54} -3136.00 q^{56} -46874.0i q^{57} +20004.0i q^{58} -5724.00 q^{59} -36112.0 q^{61} +22784.0i q^{62} +14014.0i q^{63} -4096.00 q^{64} +51060.0 q^{66} -66104.0i q^{67} -23856.0i q^{68} -28290.0 q^{69} +16080.0 q^{71} +18304.0i q^{72} -80482.0i q^{73} -22408.0 q^{74} -32608.0 q^{76} -27195.0i q^{77} -22172.0i q^{78} +64147.0 q^{79} -46751.0 q^{81} -9696.00i q^{82} -106284. i q^{83} +18032.0 q^{84} -2408.00 q^{86} -115023. i q^{87} -35520.0i q^{88} +71676.0 q^{89} -11809.0 q^{91} +19680.0i q^{92} -131008. i q^{93} -92652.0 q^{94} +23552.0 q^{96} -151025. i q^{97} -9604.00i q^{98} -158730. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} + 184 q^{6} - 572 q^{9}+O(q^{10}) $ 2 * q - 32 * q^4 + 184 * q^6 - 572 * q^9	$ 2 q - 32 q^{4} + 184 q^{6} - 572 q^{9} + 1110 q^{11} + 392 q^{14} + 512 q^{16} + 4076 q^{19} - 2254 q^{21} - 2944 q^{24} + 1928 q^{26} + 10002 q^{29} + 11392 q^{31} - 11928 q^{34} + 9152 q^{36} - 11086 q^{39} - 4848 q^{41} - 17760 q^{44} + 9840 q^{46} - 4802 q^{49} + 68586 q^{51} - 7912 q^{54} - 6272 q^{56} - 11448 q^{59} - 72224 q^{61} - 8192 q^{64} + 102120 q^{66} - 56580 q^{69} + 32160 q^{71} - 44816 q^{74} - 65216 q^{76} + 128294 q^{79} - 93502 q^{81} + 36064 q^{84} - 4816 q^{86} + 143352 q^{89} - 23618 q^{91} - 185304 q^{94} + 47104 q^{96} - 317460 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 + 184 * q^6 - 572 * q^9 + 1110 * q^11 + 392 * q^14 + 512 * q^16 + 4076 * q^19 - 2254 * q^21 - 2944 * q^24 + 1928 * q^26 + 10002 * q^29 + 11392 * q^31 - 11928 * q^34 + 9152 * q^36 - 11086 * q^39 - 4848 * q^41 - 17760 * q^44 + 9840 * q^46 - 4802 * q^49 + 68586 * q^51 - 7912 * q^54 - 6272 * q^56 - 11448 * q^59 - 72224 * q^61 - 8192 * q^64 + 102120 * q^66 - 56580 * q^69 + 32160 * q^71 - 44816 * q^74 - 65216 * q^76 + 128294 * q^79 - 93502 * q^81 + 36064 * q^84 - 4816 * q^86 + 143352 * q^89 - 23618 * q^91 - 185304 * q^94 + 47104 * q^96 - 317460 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	4.00000i	0.707107i
$2$	4.00000i	0.707107i
$3$	− 23.0000i	− 1.47545i	−0.675101	−	0.737725i	$-0.735900\pi$
$3$	− 23.0000i	− 1.47545i	0.675101	−	0.737725i	$-0.264100\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	92.0000	1.04330
$7$	− 49.0000i	− 0.377964i
$7$	− 49.0000i	− 0.377964i
$8$	− 64.0000i	− 0.353553i
$9$	−286.000	−1.17695
$10$	0	0
$11$	555.000	1.38297	0.691483	−	0.722393i	$-0.256958\pi$
$11$	555.000	1.38297	0.691483	+	0.722393i	$0.256958\pi$
$12$	368.000i	0.737725i
$13$	− 241.000i	− 0.395511i	−0.980251	−	0.197756i	$-0.936635\pi$
$13$	− 241.000i	− 0.395511i	0.980251	−	0.197756i	$-0.0633652\pi$
$14$	196.000	0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	1491.00i	1.25128i	0.780111	+	0.625641i	$0.215163\pi$
$17$	1491.00i	1.25128i	−0.780111	+	0.625641i	$0.784837\pi$
$18$	− 1144.00i	− 0.832233i
$19$	2038.00	1.29515	0.647575	−	0.762002i	$-0.275783\pi$
$19$	2038.00	1.29515	0.647575	+	0.762002i	$0.275783\pi$
$20$	0	0
$21$	−1127.00	−0.557668
$22$	2220.00i	0.977904i
$23$	− 1230.00i	− 0.484826i	−0.970173	−	0.242413i	$-0.922061\pi$
$23$	− 1230.00i	− 0.484826i	0.970173	−	0.242413i	$-0.0779388\pi$
$24$	−1472.00	−0.521651
$25$	0	0
$26$	964.000	0.279669
$27$	989.000i	0.261088i
$28$	784.000i	0.188982i
$29$	5001.00	1.10424	0.552118	−	0.833766i	$-0.313820\pi$
$29$	5001.00	1.10424	0.552118	+	0.833766i	$0.313820\pi$
$30$	0	0
$31$	5696.00	1.06455	0.532275	−	0.846572i	$-0.321337\pi$
$31$	5696.00	1.06455	0.532275	+	0.846572i	$0.321337\pi$
$32$	1024.00i	0.176777i
$33$	− 12765.0i	− 2.04050i
$34$	−5964.00	−0.884790
$35$	0	0
$36$	4576.00	0.588477
$37$	5602.00i	0.672727i	0.941732	+	0.336363i	$0.109197\pi$
$37$	5602.00i	0.672727i	−0.941732	+	0.336363i	$0.890803\pi$
$38$	8152.00i	0.915810i
$39$	−5543.00	−0.583557
$40$	0	0
$41$	−2424.00	−0.225202	−0.112601	−	0.993640i	$-0.535918\pi$
$41$	−2424.00	−0.225202	−0.112601	+	0.993640i	$0.535918\pi$
$42$	− 4508.00i	− 0.394331i
$43$	602.000i	0.0496507i	0.999692	+	0.0248253i	$0.00790297\pi$
$43$	602.000i	0.0496507i	−0.999692	+	0.0248253i	$0.992097\pi$
$44$	−8880.00	−0.691483
$45$	0	0
$46$	4920.00	0.342823
$47$	23163.0i	1.52950i	0.644326	+	0.764751i	$0.277138\pi$
$47$	23163.0i	1.52950i	−0.644326	+	0.764751i	$0.722862\pi$
$48$	− 5888.00i	− 0.368863i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	34293.0	1.84621
$52$	3856.00i	0.197756i
$53$	− 25296.0i	− 1.23698i	−0.785793	−	0.618489i	$-0.787745\pi$
$53$	− 25296.0i	− 1.23698i	0.785793	−	0.618489i	$-0.212255\pi$
$54$	−3956.00	−0.184617
$55$	0	0
$56$	−3136.00	−0.133631
$57$	− 46874.0i	− 1.91093i
$58$	20004.0i	0.780813i
$59$	−5724.00	−0.214077	−0.107038	−	0.994255i	$-0.534137\pi$
$59$	−5724.00	−0.214077	−0.107038	+	0.994255i	$0.534137\pi$
$60$	0	0
$61$	−36112.0	−1.24259	−0.621294	−	0.783578i	$-0.713393\pi$
$61$	−36112.0	−1.24259	−0.621294	+	0.783578i	$0.713393\pi$
$62$	22784.0i	0.752750i
$63$	14014.0i	0.444847i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	51060.0	1.44285
$67$	− 66104.0i	− 1.79904i	−0.436880	−	0.899520i	$-0.643917\pi$
$67$	− 66104.0i	− 1.79904i	0.436880	−	0.899520i	$-0.356083\pi$
$68$	− 23856.0i	− 0.625641i
$69$	−28290.0	−0.715336
$70$	0	0
$71$	16080.0	0.378565	0.189282	−	0.981923i	$-0.439384\pi$
$71$	16080.0	0.378565	0.189282	+	0.981923i	$0.439384\pi$
$72$	18304.0i	0.416116i
$73$	− 80482.0i	− 1.76763i	−0.467836	−	0.883816i	$-0.654966\pi$
$73$	− 80482.0i	− 1.76763i	0.467836	−	0.883816i	$-0.345034\pi$
$74$	−22408.0	−0.475690
$75$	0	0
$76$	−32608.0	−0.647575
$77$	− 27195.0i	− 0.522712i
$78$	− 22172.0i	− 0.412637i
$79$	64147.0	1.15640	0.578201	−	0.815895i	$-0.303755\pi$
$79$	64147.0	1.15640	0.578201	+	0.815895i	$0.303755\pi$
$80$	0	0
$81$	−46751.0	−0.791732
$82$	− 9696.00i	− 0.159242i
$83$	− 106284.i	− 1.69345i	−0.532030	−	0.846726i	$-0.678571\pi$
$83$	− 106284.i	− 1.69345i	0.532030	−	0.846726i	$-0.321429\pi$
$84$	18032.0	0.278834
$85$	0	0
$86$	−2408.00	−0.0351083
$87$	− 115023.i	− 1.62925i
$88$	− 35520.0i	− 0.488952i
$89$	71676.0	0.959177	0.479588	−	0.877494i	$-0.340786\pi$
$89$	71676.0	0.959177	0.479588	+	0.877494i	$0.340786\pi$
$90$	0	0
$91$	−11809.0	−0.149489
$92$	19680.0i	0.242413i
$93$	− 131008.i	− 1.57069i
$94$	−92652.0	−1.08152
$95$	0	0
$96$	23552.0	0.260825
$97$	− 151025.i	− 1.62974i	−0.579641	−	0.814872i	$-0.696807\pi$
$97$	− 151025.i	− 1.62974i	0.579641	−	0.814872i	$-0.303193\pi$
$98$	− 9604.00i	− 0.101015i
$99$	−158730.	−1.62769
$100$	0	0
$101$	−57150.0	−0.557459	−0.278729	−	0.960370i	$-0.589913\pi$
$101$	−57150.0	−0.557459	−0.278729	+	0.960370i	$0.589913\pi$
$102$	137172.i	1.30546i
$103$	115889.i	1.07634i	0.842837	+	0.538170i	$0.180884\pi$
$103$	115889.i	1.07634i	−0.842837	+	0.538170i	$0.819116\pi$
$104$	−15424.0	−0.139834
$105$	0	0
$106$	101184.	0.874676
$107$	137862.i	1.16409i	0.813158	+	0.582043i	$0.197747\pi$
$107$	137862.i	1.16409i	−0.813158	+	0.582043i	$0.802253\pi$
$108$	− 15824.0i	− 0.130544i
$109$	−88397.0	−0.712642	−0.356321	−	0.934364i	$-0.615969\pi$
$109$	−88397.0	−0.712642	−0.356321	+	0.934364i	$0.615969\pi$
$110$	0	0
$111$	128846.	0.992575
$112$	− 12544.0i	− 0.0944911i
$113$	205554.i	1.51436i	0.653205	+	0.757181i	$0.273424\pi$
$113$	205554.i	1.51436i	−0.653205	+	0.757181i	$0.726576\pi$
$114$	187496.	1.35123
$115$	0	0
$116$	−80016.0	−0.552118
$117$	68926.0i	0.465499i
$118$	− 22896.0i	− 0.151375i
$119$	73059.0	0.472940
$120$	0	0
$121$	146974.	0.912593
$122$	− 144448.i	− 0.878642i
$123$	55752.0i	0.332275i
$124$	−91136.0	−0.532275
$125$	0	0
$126$	−56056.0	−0.314554
$127$	− 250916.i	− 1.38044i	−0.723597	−	0.690222i	$-0.757513\pi$
$127$	− 250916.i	− 1.38044i	0.723597	−	0.690222i	$-0.242487\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	13846.0	0.0732572
$130$	0	0
$131$	−52122.0	−0.265365	−0.132682	−	0.991159i	$-0.542359\pi$
$131$	−52122.0	−0.265365	−0.132682	+	0.991159i	$0.542359\pi$
$132$	204240.i	1.02025i
$133$	− 99862.0i	− 0.489521i
$134$	264416.	1.27211
$135$	0	0
$136$	95424.0	0.442395
$137$	135468.i	0.616645i	0.951282	+	0.308323i	$0.0997676\pi$
$137$	135468.i	0.616645i	−0.951282	+	0.308323i	$0.900232\pi$
$138$	− 113160.i	− 0.505819i
$139$	349486.	1.53424	0.767119	−	0.641505i	$-0.221690\pi$
$139$	349486.	1.53424	0.767119	+	0.641505i	$0.221690\pi$
$140$	0	0
$141$	532749.	2.25671
$142$	64320.0i	0.267686i
$143$	− 133755.i	− 0.546978i
$144$	−73216.0	−0.294239
$145$	0	0
$146$	321928.	1.24990
$147$	55223.0i	0.210779i
$148$	− 89632.0i	− 0.336363i
$149$	−176082.	−0.649754	−0.324877	−	0.945756i	$-0.605323\pi$
$149$	−176082.	−0.649754	−0.324877	+	0.945756i	$0.605323\pi$
$150$	0	0
$151$	383333.	1.36815	0.684075	−	0.729411i	$-0.260206\pi$
$151$	383333.	1.36815	0.684075	+	0.729411i	$0.260206\pi$
$152$	− 130432.i	− 0.457905i
$153$	− 426426.i	− 1.47270i
$154$	108780.	0.369613
$155$	0	0
$156$	88688.0	0.291779
$157$	− 345914.i	− 1.12000i	−0.828492	−	0.560001i	$-0.810801\pi$
$157$	− 345914.i	− 1.12000i	0.828492	−	0.560001i	$-0.189199\pi$
$158$	256588.i	0.817699i
$159$	−581808.	−1.82510
$160$	0	0
$161$	−60270.0	−0.183247
$162$	− 187004.i	− 0.559839i
$163$	91586.0i	0.269998i	0.990846	+	0.134999i	$0.0431031\pi$
$163$	91586.0i	0.269998i	−0.990846	+	0.134999i	$0.956897\pi$
$164$	38784.0	0.112601
$165$	0	0
$166$	425136.	1.19745
$167$	− 38097.0i	− 0.105706i	−0.998602	−	0.0528530i	$-0.983169\pi$
$167$	− 38097.0i	− 0.105706i	0.998602	−	0.0528530i	$-0.0168315\pi$
$168$	72128.0i	0.197165i
$169$	313212.	0.843571
$170$	0	0
$171$	−582868.	−1.52433
$172$	− 9632.00i	− 0.0248253i
$173$	− 541443.i	− 1.37543i	−0.725982	−	0.687713i	$-0.758615\pi$
$173$	− 541443.i	− 1.37543i	0.725982	−	0.687713i	$-0.241385\pi$
$174$	460092.	1.15205
$175$	0	0
$176$	142080.	0.345741
$177$	131652.i	0.315860i
$178$	286704.i	0.678241i
$179$	−166188.	−0.387674	−0.193837	−	0.981034i	$-0.562093\pi$
$179$	−166188.	−0.387674	−0.193837	+	0.981034i	$0.562093\pi$
$180$	0	0
$181$	−197320.	−0.447687	−0.223844	−	0.974625i	$-0.571860\pi$
$181$	−197320.	−0.447687	−0.223844	+	0.974625i	$0.571860\pi$
$182$	− 47236.0i	− 0.105705i
$183$	830576.i	1.83338i
$184$	−78720.0	−0.171412
$185$	0	0
$186$	524032.	1.11065
$187$	827505.i	1.73048i
$188$	− 370608.i	− 0.764751i
$189$	48461.0	0.0986820
$190$	0	0
$191$	−337221.	−0.668854	−0.334427	−	0.942422i	$-0.608543\pi$
$191$	−337221.	−0.668854	−0.334427	+	0.942422i	$0.608543\pi$
$192$	94208.0i	0.184431i
$193$	260516.i	0.503432i	0.967801	+	0.251716i	$0.0809949\pi$
$193$	260516.i	0.503432i	−0.967801	+	0.251716i	$0.919005\pi$
$194$	604100.	1.15240
$195$	0	0
$196$	38416.0	0.0714286
$197$	409212.i	0.751247i	0.926772	+	0.375624i	$0.122571\pi$
$197$	409212.i	0.751247i	−0.926772	+	0.375624i	$0.877429\pi$
$198$	− 634920.i	− 1.15095i
$199$	−300980.	−0.538772	−0.269386	−	0.963032i	$-0.586821\pi$
$199$	−300980.	−0.538772	−0.269386	+	0.963032i	$0.586821\pi$
$200$	0	0
$201$	−1.52039e6	−2.65439
$202$	− 228600.i	− 0.394183i
$203$	− 245049.i	− 0.417362i
$204$	−548688.	−0.923103
$205$	0	0
$206$	−463556.	−0.761087
$207$	351780.i	0.570618i
$208$	− 61696.0i	− 0.0988778i
$209$	1.13109e6	1.79115
$210$	0	0
$211$	−1.22618e6	−1.89604	−0.948021	−	0.318209i	$-0.896919\pi$
$211$	−1.22618e6	−1.89604	−0.948021	+	0.318209i	$0.896919\pi$
$212$	404736.i	0.618489i
$213$	− 369840.i	− 0.558554i
$214$	−551448.	−0.823133
$215$	0	0
$216$	63296.0	0.0923085
$217$	− 279104.i	− 0.402362i
$218$	− 353588.i	− 0.503914i
$219$	−1.85109e6	−2.60805
$220$	0	0
$221$	359331.	0.494896
$222$	515384.i	0.701857i
$223$	621257.i	0.836583i	0.908313	+	0.418292i	$0.137371\pi$
$223$	621257.i	0.836583i	−0.908313	+	0.418292i	$0.862629\pi$
$224$	50176.0	0.0668153
$225$	0	0
$226$	−822216.	−1.07082
$227$	− 1.29768e6i	− 1.67148i	−0.549123	−	0.835742i	$-0.685038\pi$
$227$	− 1.29768e6i	− 1.67148i	0.549123	−	0.835742i	$-0.314962\pi$
$228$	749984.i	0.955465i
$229$	124264.	0.156587	0.0782937	−	0.996930i	$-0.475053\pi$
$229$	124264.	0.156587	0.0782937	+	0.996930i	$0.475053\pi$
$230$	0	0
$231$	−625485.	−0.771235
$232$	− 320064.i	− 0.390406i
$233$	1.08742e6i	1.31222i	0.754666	+	0.656109i	$0.227799\pi$
$233$	1.08742e6i	1.31222i	−0.754666	+	0.656109i	$0.772201\pi$
$234$	−275704.	−0.329157
$235$	0	0
$236$	91584.0	0.107038
$237$	− 1.47538e6i	− 1.70621i
$238$	292236.i	0.334419i
$239$	545631.	0.617880	0.308940	−	0.951081i	$-0.400026\pi$
$239$	545631.	0.617880	0.308940	+	0.951081i	$0.400026\pi$
$240$	0	0
$241$	811310.	0.899796	0.449898	−	0.893080i	$-0.351460\pi$
$241$	811310.	0.899796	0.449898	+	0.893080i	$0.351460\pi$
$242$	587896.i	0.645301i
$243$	1.31560e6i	1.42925i
$244$	577792.	0.621294
$245$	0	0
$246$	−223008.	−0.234954
$247$	− 491158.i	− 0.512246i
$248$	− 364544.i	− 0.376375i
$249$	−2.44453e6	−2.49860
$250$	0	0
$251$	−897738.	−0.899426	−0.449713	−	0.893173i	$-0.648474\pi$
$251$	−897738.	−0.899426	−0.449713	+	0.893173i	$0.648474\pi$
$252$	− 224224.i	− 0.222424i
$253$	− 682650.i	− 0.670497i
$254$	1.00366e6	0.976122
$255$	0	0
$256$	65536.0	0.0625000
$257$	594678.i	0.561628i	0.959762	+	0.280814i	$0.0906045\pi$
$257$	594678.i	0.561628i	−0.959762	+	0.280814i	$0.909396\pi$
$258$	55384.0i	0.0518006i
$259$	274498.	0.254267
$260$	0	0
$261$	−1.43029e6	−1.29964
$262$	− 208488.i	− 0.187641i
$263$	1.02837e6i	0.916769i	0.888754	+	0.458385i	$0.151572\pi$
$263$	1.02837e6i	0.916769i	−0.888754	+	0.458385i	$0.848428\pi$
$264$	−816960.	−0.721425
$265$	0	0
$266$	399448.	0.346143
$267$	− 1.64855e6i	− 1.41522i
$268$	1.05766e6i	0.899520i
$269$	1.24390e6	1.04811	0.524053	−	0.851685i	$-0.324419\pi$
$269$	1.24390e6	1.04811	0.524053	+	0.851685i	$0.324419\pi$
$270$	0	0
$271$	737624.	0.610115	0.305058	−	0.952334i	$-0.401324\pi$
$271$	737624.	0.610115	0.305058	+	0.952334i	$0.401324\pi$
$272$	381696.i	0.312821i
$273$	271607.i	0.220564i
$274$	−541872.	−0.436034
$275$	0	0
$276$	452640.	0.357668
$277$	2.20063e6i	1.72325i	0.507548	+	0.861624i	$0.330552\pi$
$277$	2.20063e6i	1.72325i	−0.507548	+	0.861624i	$0.669448\pi$
$278$	1.39794e6i	1.08487i
$279$	−1.62906e6	−1.25293
$280$	0	0
$281$	173979.	0.131441	0.0657205	−	0.997838i	$-0.479065\pi$
$281$	173979.	0.131441	0.0657205	+	0.997838i	$0.479065\pi$
$282$	2.13100e6i	1.59573i
$283$	− 551053.i	− 0.409004i	−0.978866	−	0.204502i	$-0.934443\pi$
$283$	− 551053.i	− 0.409004i	0.978866	−	0.204502i	$-0.0655574\pi$
$284$	−257280.	−0.189282
$285$	0	0
$286$	535020.	0.386772
$287$	118776.i	0.0851185i
$288$	− 292864.i	− 0.208058i
$289$	−803224.	−0.565708
$290$	0	0
$291$	−3.47358e6	−2.40461
$292$	1.28771e6i	0.883816i
$293$	− 1.67512e6i	− 1.13993i	−0.821670	−	0.569963i	$-0.806958\pi$
$293$	− 1.67512e6i	− 1.13993i	0.821670	−	0.569963i	$-0.193042\pi$
$294$	−220892.	−0.149043
$295$	0	0
$296$	358528.	0.237845
$297$	548895.i	0.361076i
$298$	− 704328.i	− 0.459446i
$299$	−296430.	−0.191754
$300$	0	0
$301$	29498.0	0.0187662
$302$	1.53333e6i	0.967428i
$303$	1.31445e6i	0.822503i
$304$	521728.	0.323788
$305$	0	0
$306$	1.70570e6	1.04136
$307$	− 2.33060e6i	− 1.41131i	−0.708556	−	0.705655i	$-0.750653\pi$
$307$	− 2.33060e6i	− 1.41131i	0.708556	−	0.705655i	$-0.249347\pi$
$308$	435120.i	0.261356i
$309$	2.66545e6	1.58809
$310$	0	0
$311$	706266.	0.414064	0.207032	−	0.978334i	$-0.433620\pi$
$311$	706266.	0.414064	0.207032	+	0.978334i	$0.433620\pi$
$312$	354752.i	0.206319i
$313$	− 183565.i	− 0.105908i	−0.998597	−	0.0529540i	$-0.983136\pi$
$313$	− 183565.i	− 0.105908i	0.998597	−	0.0529540i	$-0.0168637\pi$
$314$	1.38366e6	0.791961
$315$	0	0
$316$	−1.02635e6	−0.578201
$317$	− 2.70665e6i	− 1.51281i	−0.654103	−	0.756405i	$-0.726954\pi$
$317$	− 2.70665e6i	− 1.51281i	0.654103	−	0.756405i	$-0.273046\pi$
$318$	− 2.32723e6i	− 1.29054i
$319$	2.77556e6	1.52712
$320$	0	0
$321$	3.17083e6	1.71755
$322$	− 241080.i	− 0.129575i
$323$	3.03866e6i	1.62060i
$324$	748016.	0.395866
$325$	0	0
$326$	−366344.	−0.190917
$327$	2.03313e6i	1.05147i
$328$	155136.i	0.0796211i
$329$	1.13499e6	0.578098
$330$	0	0
$331$	−2.14337e6	−1.07529	−0.537647	−	0.843170i	$-0.680687\pi$
$331$	−2.14337e6	−1.07529	−0.537647	+	0.843170i	$0.680687\pi$
$332$	1.70054e6i	0.846726i
$333$	− 1.60217e6i	− 0.791769i
$334$	152388.	0.0747454
$335$	0	0
$336$	−288512.	−0.139417
$337$	− 655346.i	− 0.314337i	−0.987572	−	0.157169i	$-0.949763\pi$
$337$	− 655346.i	− 0.314337i	0.987572	−	0.157169i	$-0.0502366\pi$
$338$	1.25285e6i	0.596495i
$339$	4.72774e6	2.23437
$340$	0	0
$341$	3.16128e6	1.47223
$342$	− 2.33147e6i	− 1.07787i
$343$	117649.i	0.0539949i
$344$	38528.0	0.0175542
$345$	0	0
$346$	2.16577e6	0.972574
$347$	4.22275e6i	1.88266i	0.337491	+	0.941329i	$0.390422\pi$
$347$	4.22275e6i	1.88266i	−0.337491	+	0.941329i	$0.609578\pi$
$348$	1.84037e6i	0.814623i
$349$	−3.01710e6	−1.32595	−0.662974	−	0.748643i	$-0.730706\pi$
$349$	−3.01710e6	−1.32595	−0.662974	+	0.748643i	$0.730706\pi$
$350$	0	0
$351$	238349.	0.103263
$352$	568320.i	0.244476i
$353$	2.25258e6i	0.962150i	0.876679	+	0.481075i	$0.159754\pi$
$353$	2.25258e6i	0.962150i	−0.876679	+	0.481075i	$0.840246\pi$
$354$	−526608.	−0.223347
$355$	0	0
$356$	−1.14682e6	−0.479588
$357$	− 1.68036e6i	− 0.697800i
$358$	− 664752.i	− 0.274127i
$359$	1.83950e6	0.753294	0.376647	−	0.926357i	$-0.377077\pi$
$359$	1.83950e6	0.753294	0.376647	+	0.926357i	$0.377077\pi$
$360$	0	0
$361$	1.67735e6	0.677414
$362$	− 789280.i	− 0.316563i
$363$	− 3.38040e6i	− 1.34649i
$364$	188944.	0.0747446
$365$	0	0
$366$	−3.32230e6	−1.29639
$367$	1.68832e6i	0.654320i	0.944969	+	0.327160i	$0.106092\pi$
$367$	1.68832e6i	0.654320i	−0.944969	+	0.327160i	$0.893908\pi$
$368$	− 314880.i	− 0.121206i
$369$	693264.	0.265053
$370$	0	0
$371$	−1.23950e6	−0.467534
$372$	2.09613e6i	0.785345i
$373$	1.81212e6i	0.674394i	0.941434	+	0.337197i	$0.109479\pi$
$373$	1.81212e6i	0.674394i	−0.941434	+	0.337197i	$0.890521\pi$
$374$	−3.31002e6	−1.22363
$375$	0	0
$376$	1.48243e6	0.540761
$377$	− 1.20524e6i	− 0.436738i
$378$	193844.i	0.0697787i
$379$	4.76708e6	1.70472	0.852362	−	0.522952i	$-0.175169\pi$
$379$	4.76708e6	1.70472	0.852362	+	0.522952i	$0.175169\pi$
$380$	0	0
$381$	−5.77107e6	−2.03678
$382$	− 1.34888e6i	− 0.472951i
$383$	− 69996.0i	− 0.0243824i	−0.999926	−	0.0121912i	$-0.996119\pi$
$383$	− 69996.0i	− 0.0243824i	0.999926	−	0.0121912i	$-0.00388067\pi$
$384$	−376832.	−0.130413
$385$	0	0
$386$	−1.04206e6	−0.355980
$387$	− 172172.i	− 0.0584366i
$388$	2.41640e6i	0.814872i
$389$	−3.98895e6	−1.33655	−0.668275	−	0.743915i	$-0.732967\pi$
$389$	−3.98895e6	−1.33655	−0.668275	+	0.743915i	$0.732967\pi$
$390$	0	0
$391$	1.83393e6	0.606654
$392$	153664.i	0.0505076i
$393$	1.19881e6i	0.391532i
$394$	−1.63685e6	−0.531212
$395$	0	0
$396$	2.53968e6	0.813844
$397$	3.05904e6i	0.974110i	0.873371	+	0.487055i	$0.161929\pi$
$397$	3.05904e6i	0.974110i	−0.873371	+	0.487055i	$0.838071\pi$
$398$	− 1.20392e6i	− 0.380969i
$399$	−2.29683e6	−0.722264
$400$	0	0
$401$	4.30794e6	1.33785	0.668927	−	0.743329i	$-0.266754\pi$
$401$	4.30794e6	1.33785	0.668927	+	0.743329i	$0.266754\pi$
$402$	− 6.08157e6i	− 1.87694i
$403$	− 1.37274e6i	− 0.421041i
$404$	914400.	0.278729
$405$	0	0
$406$	980196.	0.295119
$407$	3.10911e6i	0.930358i
$408$	− 2.19475e6i	− 0.652732i
$409$	239206.	0.0707072	0.0353536	−	0.999375i	$-0.488744\pi$
$409$	239206.	0.0707072	0.0353536	+	0.999375i	$0.488744\pi$
$410$	0	0
$411$	3.11576e6	0.909829
$412$	− 1.85422e6i	− 0.538170i
$413$	280476.i	0.0809134i
$414$	−1.40712e6	−0.403488
$415$	0	0
$416$	246784.	0.0699171
$417$	− 8.03818e6i	− 2.26369i
$418$	4.52436e6i	1.26653i
$419$	−4.63462e6	−1.28967	−0.644835	−	0.764322i	$-0.723074\pi$
$419$	−4.63462e6	−1.28967	−0.644835	+	0.764322i	$0.723074\pi$
$420$	0	0
$421$	−2.10108e6	−0.577745	−0.288873	−	0.957368i	$-0.593280\pi$
$421$	−2.10108e6	−0.577745	−0.288873	+	0.957368i	$0.593280\pi$
$422$	− 4.90472e6i	− 1.34070i
$423$	− 6.62462e6i	− 1.80016i
$424$	−1.61894e6	−0.437338
$425$	0	0
$426$	1.47936e6	0.394957
$427$	1.76949e6i	0.469654i
$428$	− 2.20579e6i	− 0.582043i
$429$	−3.07636e6	−0.807039
$430$	0	0
$431$	1.65484e6	0.429104	0.214552	−	0.976713i	$-0.431171\pi$
$431$	1.65484e6	0.429104	0.214552	+	0.976713i	$0.431171\pi$
$432$	253184.i	0.0652720i
$433$	− 1.84031e6i	− 0.471705i	−0.971789	−	0.235852i	$-0.924212\pi$
$433$	− 1.84031e6i	− 0.471705i	0.971789	−	0.235852i	$-0.0757882\pi$
$434$	1.11642e6	0.284513
$435$	0	0
$436$	1.41435e6	0.356321
$437$	− 2.50674e6i	− 0.627922i
$438$	− 7.40434e6i	− 1.84417i
$439$	−5.83684e6	−1.44549	−0.722747	−	0.691113i	$-0.757121\pi$
$439$	−5.83684e6	−1.44549	−0.722747	+	0.691113i	$0.757121\pi$
$440$	0	0
$441$	686686.	0.168136
$442$	1.43732e6i	0.349944i
$443$	1.19704e6i	0.289801i	0.989446	+	0.144901i	$0.0462862\pi$
$443$	1.19704e6i	0.289801i	−0.989446	+	0.144901i	$0.953714\pi$
$444$	−2.06154e6	−0.496288
$445$	0	0
$446$	−2.48503e6	−0.591554
$447$	4.04989e6i	0.958681i
$448$	200704.i	0.0472456i
$449$	3.42570e6	0.801924	0.400962	−	0.916095i	$-0.368676\pi$
$449$	3.42570e6	0.801924	0.400962	+	0.916095i	$0.368676\pi$
$450$	0	0
$451$	−1.34532e6	−0.311447
$452$	− 3.28886e6i	− 0.757181i
$453$	− 8.81666e6i	− 2.01864i
$454$	5.19071e6	1.18192
$455$	0	0
$456$	−2.99994e6	−0.675616
$457$	− 5.29742e6i	− 1.18652i	−0.805012	−	0.593258i	$-0.797841\pi$
$457$	− 5.29742e6i	− 1.18652i	0.805012	−	0.593258i	$-0.202159\pi$
$458$	497056.i	0.110724i
$459$	−1.47460e6	−0.326695
$460$	0	0
$461$	8.87731e6	1.94549	0.972745	−	0.231876i	$-0.0744863\pi$
$461$	8.87731e6	1.94549	0.972745	+	0.231876i	$0.0744863\pi$
$462$	− 2.50194e6i	− 0.545346i
$463$	− 2.17475e6i	− 0.471473i	−0.971817	−	0.235737i	$-0.924250\pi$
$463$	− 2.17475e6i	− 0.471473i	0.971817	−	0.235737i	$-0.0757503\pi$
$464$	1.28026e6	0.276059
$465$	0	0
$466$	−4.34966e6	−0.927878
$467$	378969.i	0.0804103i	0.999191	+	0.0402051i	$0.0128011\pi$
$467$	378969.i	0.0804103i	−0.999191	+	0.0402051i	$0.987199\pi$
$468$	− 1.10282e6i	− 0.232749i
$469$	−3.23910e6	−0.679973
$470$	0	0
$471$	−7.95602e6	−1.65251
$472$	366336.i	0.0756876i
$473$	334110.i	0.0686652i
$474$	5.90152e6	1.20647
$475$	0	0
$476$	−1.16894e6	−0.236470
$477$	7.23466e6i	1.45587i
$478$	2.18252e6i	0.436907i
$479$	−1.88489e6	−0.375360	−0.187680	−	0.982230i	$-0.560097\pi$
$479$	−1.88489e6	−0.375360	−0.187680	+	0.982230i	$0.560097\pi$
$480$	0	0
$481$	1.35008e6	0.266071
$482$	3.24524e6i	0.636252i
$483$	1.38621e6i	0.270372i
$484$	−2.35158e6	−0.456296
$485$	0	0
$486$	−5.26240e6	−1.01063
$487$	− 3.67689e6i	− 0.702518i	−0.936278	−	0.351259i	$-0.885754\pi$
$487$	− 3.67689e6i	− 0.702518i	0.936278	−	0.351259i	$-0.114246\pi$
$488$	2.31117e6i	0.439321i
$489$	2.10648e6	0.398368
$490$	0	0
$491$	9.54015e6	1.78588	0.892939	−	0.450178i	$-0.148640\pi$
$491$	9.54015e6	1.78588	0.892939	+	0.450178i	$0.148640\pi$
$492$	− 892032.i	− 0.166138i
$493$	7.45649e6i	1.38171i
$494$	1.96463e6	0.362213
$495$	0	0
$496$	1.45818e6	0.266137
$497$	− 787920.i	− 0.143084i
$498$	− 9.77813e6i	− 1.76678i
$499$	−4.78243e6	−0.859800	−0.429900	−	0.902877i	$-0.641451\pi$
$499$	−4.78243e6	−0.859800	−0.429900	+	0.902877i	$0.641451\pi$
$500$	0	0
$501$	−876231.	−0.155964
$502$	− 3.59095e6i	− 0.635990i
$503$	− 1.08395e7i	− 1.91024i	−0.296220	−	0.955120i	$-0.595726\pi$
$503$	− 1.08395e7i	− 1.91024i	0.296220	−	0.955120i	$-0.404274\pi$
$504$	896896.	0.157277
$505$	0	0
$506$	2.73060e6	0.474113
$507$	− 7.20388e6i	− 1.24465i
$508$	4.01466e6i	0.690222i
$509$	7.64177e6	1.30737	0.653687	−	0.756765i	$-0.273221\pi$
$509$	7.64177e6	1.30737	0.653687	+	0.756765i	$0.273221\pi$
$510$	0	0
$511$	−3.94362e6	−0.668102
$512$	262144.i	0.0441942i
$513$	2.01558e6i	0.338148i
$514$	−2.37871e6	−0.397131
$515$	0	0
$516$	−221536.	−0.0366286
$517$	1.28555e7i	2.11525i
$518$	1.09799e6i	0.179794i
$519$	−1.24532e7	−2.02937
$520$	0	0
$521$	6.44011e6	1.03944	0.519719	−	0.854337i	$-0.326037\pi$
$521$	6.44011e6	1.03944	0.519719	+	0.854337i	$0.326037\pi$
$522$	− 5.72114e6i	− 0.918981i
$523$	− 4.77929e6i	− 0.764028i	−0.924157	−	0.382014i	$-0.875231\pi$
$523$	− 4.77929e6i	− 0.764028i	0.924157	−	0.382014i	$-0.124769\pi$
$524$	833952.	0.132682
$525$	0	0
$526$	−4.11348e6	−0.648254
$527$	8.49274e6i	1.33205i
$528$	− 3.26784e6i	− 0.510124i
$529$	4.92344e6	0.764944
$530$	0	0
$531$	1.63706e6	0.251959
$532$	1.59779e6i	0.244760i
$533$	584184.i	0.0890700i
$534$	6.59419e6	1.00071
$535$	0	0
$536$	−4.23066e6	−0.636057
$537$	3.82232e6i	0.571994i
$538$	4.97561e6i	0.741123i
$539$	−1.33256e6	−0.197566
$540$	0	0
$541$	−2.05678e6	−0.302130	−0.151065	−	0.988524i	$-0.548270\pi$
$541$	−2.05678e6	−0.302130	−0.151065	+	0.988524i	$0.548270\pi$
$542$	2.95050e6i	0.431417i
$543$	4.53836e6i	0.660540i
$544$	−1.52678e6	−0.221198
$545$	0	0
$546$	−1.08643e6	−0.155962
$547$	1.20189e7i	1.71750i	0.512393	+	0.858751i	$0.328759\pi$
$547$	1.20189e7i	1.71750i	−0.512393	+	0.858751i	$0.671241\pi$
$548$	− 2.16749e6i	− 0.308323i
$549$	1.03280e7	1.46247
$550$	0	0
$551$	1.01920e7	1.43015
$552$	1.81056e6i	0.252910i
$553$	− 3.14320e6i	− 0.437079i
$554$	−8.80252e6	−1.21852
$555$	0	0
$556$	−5.59178e6	−0.767119
$557$	− 8.69942e6i	− 1.18810i	−0.804429	−	0.594049i	$-0.797528\pi$
$557$	− 8.69942e6i	− 1.18810i	0.804429	−	0.594049i	$-0.202472\pi$
$558$	− 6.51622e6i	− 0.885953i
$559$	145082.	0.0196374
$560$	0	0
$561$	1.90326e7	2.55324
$562$	695916.i	0.0929429i
$563$	− 7.35942e6i	− 0.978527i	−0.872136	−	0.489263i	$-0.837266\pi$
$563$	− 7.35942e6i	− 0.978527i	0.872136	−	0.489263i	$-0.162734\pi$
$564$	−8.52398e6	−1.12835
$565$	0	0
$566$	2.20421e6	0.289209
$567$	2.29080e6i	0.299247i
$568$	− 1.02912e6i	− 0.133843i
$569$	7.50029e6	0.971175	0.485588	−	0.874188i	$-0.338606\pi$
$569$	7.50029e6	0.971175	0.485588	+	0.874188i	$0.338606\pi$
$570$	0	0
$571$	−2.22879e6	−0.286074	−0.143037	−	0.989717i	$-0.545687\pi$
$571$	−2.22879e6	−0.286074	−0.143037	+	0.989717i	$0.545687\pi$
$572$	2.14008e6i	0.273489i
$573$	7.75608e6i	0.986861i
$574$	−475104.	−0.0601879
$575$	0	0
$576$	1.17146e6	0.147119
$577$	5.10946e6i	0.638903i	0.947603	+	0.319452i	$0.103499\pi$
$577$	5.10946e6i	0.638903i	−0.947603	+	0.319452i	$0.896501\pi$
$578$	− 3.21290e6i	− 0.400016i
$579$	5.99187e6	0.742790
$580$	0	0
$581$	−5.20792e6	−0.640064
$582$	− 1.38943e7i	− 1.70031i
$583$	− 1.40393e7i	− 1.71070i
$584$	−5.15085e6	−0.624952
$585$	0	0
$586$	6.70048e6	0.806049
$587$	− 1.10646e7i	− 1.32537i	−0.748896	−	0.662687i	$-0.769416\pi$
$587$	− 1.10646e7i	− 1.32537i	0.748896	−	0.662687i	$-0.230584\pi$
$588$	− 883568.i	− 0.105389i
$589$	1.16084e7	1.37875
$590$	0	0
$591$	9.41188e6	1.10843
$592$	1.43411e6i	0.168182i
$593$	9.10043e6i	1.06274i	0.847141	+	0.531368i	$0.178322\pi$
$593$	9.10043e6i	1.06274i	−0.847141	+	0.531368i	$0.821678\pi$
$594$	−2.19558e6	−0.255319
$595$	0	0
$596$	2.81731e6	0.324877
$597$	6.92254e6i	0.794931i
$598$	− 1.18572e6i	− 0.135590i
$599$	1.28615e7	1.46462	0.732309	−	0.680973i	$-0.238443\pi$
$599$	1.28615e7	1.46462	0.732309	+	0.680973i	$0.238443\pi$
$600$	0	0
$601$	1.58163e6	0.178615	0.0893074	−	0.996004i	$-0.471535\pi$
$601$	1.58163e6	0.178615	0.0893074	+	0.996004i	$0.471535\pi$
$602$	117992.i	0.0132697i
$603$	1.89057e7i	2.11739i
$604$	−6.13333e6	−0.684075
$605$	0	0
$606$	−5.25780e6	−0.581597
$607$	688297.i	0.0758236i	0.999281	+	0.0379118i	$0.0120706\pi$
$607$	688297.i	0.0758236i	−0.999281	+	0.0379118i	$0.987929\pi$
$608$	2.08691e6i	0.228952i
$609$	−5.63613e6	−0.615797
$610$	0	0
$611$	5.58228e6	0.604935
$612$	6.82282e6i	0.736351i
$613$	− 6.02150e6i	− 0.647223i	−0.946190	−	0.323611i	$-0.895103\pi$
$613$	− 6.02150e6i	− 0.647223i	0.946190	−	0.323611i	$-0.104897\pi$
$614$	9.32241e6	0.997947
$615$	0	0
$616$	−1.74048e6	−0.184807
$617$	− 3.36137e6i	− 0.355471i	−0.984078	−	0.177735i	$-0.943123\pi$
$617$	− 3.36137e6i	− 0.355471i	0.984078	−	0.177735i	$-0.0568771\pi$
$618$	1.06618e7i	1.12295i
$619$	−1.31769e7	−1.38225	−0.691126	−	0.722734i	$-0.742885\pi$
$619$	−1.31769e7	−1.38225	−0.691126	+	0.722734i	$0.742885\pi$
$620$	0	0
$621$	1.21647e6	0.126582
$622$	2.82506e6i	0.292787i
$623$	− 3.51212e6i	− 0.362535i
$624$	−1.41901e6	−0.145889
$625$	0	0
$626$	734260.	0.0748883
$627$	− 2.60151e7i	− 2.64275i
$628$	5.53462e6i	0.560001i
$629$	−8.35258e6	−0.841771
$630$	0	0
$631$	1.26264e6	0.126243	0.0631213	−	0.998006i	$-0.479895\pi$
$631$	1.26264e6	0.126243	0.0631213	+	0.998006i	$0.479895\pi$
$632$	− 4.10541e6i	− 0.408850i
$633$	2.82021e7i	2.79752i
$634$	1.08266e7	1.06972
$635$	0	0
$636$	9.30893e6	0.912550
$637$	578641.i	0.0565016i
$638$	1.11022e7i	1.07984i
$639$	−4.59888e6	−0.445554
$640$	0	0
$641$	−1.58859e7	−1.52710	−0.763550	−	0.645749i	$-0.776545\pi$
$641$	−1.58859e7	−1.52710	−0.763550	+	0.645749i	$0.776545\pi$
$642$	1.26833e7i	1.21449i
$643$	1.80880e6i	0.172529i	0.996272	+	0.0862647i	$0.0274931\pi$
$643$	1.80880e6i	0.172529i	−0.996272	+	0.0862647i	$0.972507\pi$
$644$	964320.	0.0916234
$645$	0	0
$646$	−1.21546e7	−1.14594
$647$	− 95712.0i	− 0.00898888i	−0.999990	−	0.00449444i	$-0.998569\pi$
$647$	− 95712.0i	− 0.00898888i	0.999990	−	0.00449444i	$-0.00143063\pi$
$648$	2.99206e6i	0.279920i
$649$	−3.17682e6	−0.296061
$650$	0	0
$651$	−6.41939e6	−0.593665
$652$	− 1.46538e6i	− 0.134999i
$653$	3.06736e6i	0.281502i	0.990045	+	0.140751i	$0.0449518\pi$
$653$	3.06736e6i	0.281502i	−0.990045	+	0.140751i	$0.955048\pi$
$654$	−8.13252e6	−0.743500
$655$	0	0
$656$	−620544.	−0.0563006
$657$	2.30179e7i	2.08042i
$658$	4.53995e6i	0.408777i
$659$	1.32961e6	0.119264	0.0596321	−	0.998220i	$-0.481007\pi$
$659$	1.32961e6	0.119264	0.0596321	+	0.998220i	$0.481007\pi$
$660$	0	0
$661$	−7.37188e6	−0.656258	−0.328129	−	0.944633i	$-0.606418\pi$
$661$	−7.37188e6	−0.656258	−0.328129	+	0.944633i	$0.606418\pi$
$662$	− 8.57349e6i	− 0.760348i
$663$	− 8.26461e6i	− 0.730195i
$664$	−6.80218e6	−0.598725
$665$	0	0
$666$	6.40869e6	0.559865
$667$	− 6.15123e6i	− 0.535362i
$668$	609552.i	0.0528530i
$669$	1.42889e7	1.23434
$670$	0	0
$671$	−2.00422e7	−1.71846
$672$	− 1.15405e6i	− 0.0985827i
$673$	8.48476e6i	0.722108i	0.932545	+	0.361054i	$0.117583\pi$
$673$	8.48476e6i	0.722108i	−0.932545	+	0.361054i	$0.882417\pi$
$674$	2.62138e6	0.222270
$675$	0	0
$676$	−5.01139e6	−0.421785
$677$	4.35891e6i	0.365516i	0.983158	+	0.182758i	$0.0585025\pi$
$677$	4.35891e6i	0.365516i	−0.983158	+	0.182758i	$0.941497\pi$
$678$	1.89110e7i	1.57994i
$679$	−7.40022e6	−0.615985
$680$	0	0
$681$	−2.98466e7	−2.46619
$682$	1.26451e7i	1.04103i
$683$	− 1.58732e7i	− 1.30200i	−0.759077	−	0.651001i	$-0.774349\pi$
$683$	− 1.58732e7i	− 1.30200i	0.759077	−	0.651001i	$-0.225651\pi$
$684$	9.32589e6	0.762167
$685$	0	0
$686$	−470596.	−0.0381802
$687$	− 2.85807e6i	− 0.231037i
$688$	154112.i	0.0124127i
$689$	−6.09634e6	−0.489239
$690$	0	0
$691$	−554956.	−0.0442144	−0.0221072	−	0.999756i	$-0.507038\pi$
$691$	−554956.	−0.0442144	−0.0221072	+	0.999756i	$0.507038\pi$
$692$	8.66309e6i	0.687713i
$693$	7.77777e6i	0.615208i
$694$	−1.68910e7	−1.33124
$695$	0	0
$696$	−7.36147e6	−0.576025
$697$	− 3.61418e6i	− 0.281792i
$698$	− 1.20684e7i	− 0.937587i
$699$	2.50106e7	1.93611
$700$	0	0
$701$	−7.74720e6	−0.595456	−0.297728	−	0.954651i	$-0.596229\pi$
$701$	−7.74720e6	−0.595456	−0.297728	+	0.954651i	$0.596229\pi$
$702$	953396.i	0.0730181i
$703$	1.14169e7i	0.871282i
$704$	−2.27328e6	−0.172871
$705$	0	0
$706$	−9.01031e6	−0.680343
$707$	2.80035e6i	0.210700i
$708$	− 2.10643e6i	− 0.157930i
$709$	1.89055e7	1.41245	0.706225	−	0.707987i	$-0.250397\pi$
$709$	1.89055e7	1.41245	0.706225	+	0.707987i	$0.250397\pi$
$710$	0	0
$711$	−1.83460e7	−1.36103
$712$	− 4.58726e6i	− 0.339120i
$713$	− 7.00608e6i	− 0.516121i
$714$	6.72143e6	0.493419
$715$	0	0
$716$	2.65901e6	0.193837
$717$	− 1.25495e7i	− 0.911652i
$718$	7.35802e6i	0.532659i
$719$	1.83928e7	1.32686	0.663430	−	0.748238i	$-0.269100\pi$
$719$	1.83928e7	1.32686	0.663430	+	0.748238i	$0.269100\pi$
$720$	0	0
$721$	5.67856e6	0.406818
$722$	6.70938e6i	0.479004i
$723$	− 1.86601e7i	− 1.32761i
$724$	3.15712e6	0.223844
$725$	0	0
$726$	1.35216e7	0.952109
$727$	1.34259e7i	0.942123i	0.882100	+	0.471061i	$0.156129\pi$
$727$	1.34259e7i	0.942123i	−0.882100	+	0.471061i	$0.843871\pi$
$728$	755776.i	0.0528524i
$729$	1.88983e7	1.31706
$730$	0	0
$731$	−897582.	−0.0621270
$732$	− 1.32892e7i	− 0.916688i
$733$	1.08473e7i	0.745697i	0.927892	+	0.372848i	$0.121619\pi$
$733$	1.08473e7i	0.745697i	−0.927892	+	0.372848i	$0.878381\pi$
$734$	−6.75329e6	−0.462674
$735$	0	0
$736$	1.25952e6	0.0857059
$737$	− 3.66877e7i	− 2.48801i
$738$	2.77306e6i	0.187421i
$739$	−2.64323e7	−1.78043	−0.890214	−	0.455542i	$-0.849445\pi$
$739$	−2.64323e7	−1.78043	−0.890214	+	0.455542i	$0.849445\pi$
$740$	0	0
$741$	−1.12966e7	−0.755794
$742$	− 4.95802e6i	− 0.330596i
$743$	2.03120e7i	1.34984i	0.737893	+	0.674918i	$0.235821\pi$
$743$	2.03120e7i	1.34984i	−0.737893	+	0.674918i	$0.764179\pi$
$744$	−8.38451e6	−0.555323
$745$	0	0
$746$	−7.24846e6	−0.476869
$747$	3.03972e7i	1.99312i
$748$	− 1.32401e7i	− 0.865240i
$749$	6.75524e6	0.439983
$750$	0	0
$751$	−3.95388e6	−0.255813	−0.127907	−	0.991786i	$-0.540826\pi$
$751$	−3.95388e6	−0.255813	−0.127907	+	0.991786i	$0.540826\pi$
$752$	5.92973e6i	0.382376i
$753$	2.06480e7i	1.32706i
$754$	4.82096e6	0.308820
$755$	0	0
$756$	−775376.	−0.0493410
$757$	2.62165e7i	1.66278i	0.555688	+	0.831391i	$0.312455\pi$
$757$	2.62165e7i	1.66278i	−0.555688	+	0.831391i	$0.687545\pi$
$758$	1.90683e7i	1.20542i
$759$	−1.57010e7	−0.989285
$760$	0	0
$761$	1.14329e7	0.715638	0.357819	−	0.933791i	$-0.383521\pi$
$761$	1.14329e7	0.715638	0.357819	+	0.933791i	$0.383521\pi$
$762$	− 2.30843e7i	− 1.44022i
$763$	4.33145e6i	0.269353i
$764$	5.39554e6	0.334427
$765$	0	0
$766$	279984.	0.0172410
$767$	1.37948e6i	0.0846697i
$768$	− 1.50733e6i	− 0.0922157i
$769$	−2.37076e7	−1.44568	−0.722840	−	0.691015i	$-0.757164\pi$
$769$	−2.37076e7	−1.44568	−0.722840	+	0.691015i	$0.757164\pi$
$770$	0	0
$771$	1.36776e7	0.828655
$772$	− 4.16826e6i	− 0.251716i
$773$	− 1.24180e7i	− 0.747484i	−0.927533	−	0.373742i	$-0.878075\pi$
$773$	− 1.24180e7i	− 0.747484i	0.927533	−	0.373742i	$-0.121925\pi$
$774$	688688.	0.0413209
$775$	0	0
$776$	−9.66560e6	−0.576202
$777$	− 6.31345e6i	− 0.375158i
$778$	− 1.59558e7i	− 0.945083i
$779$	−4.94011e6	−0.291671
$780$	0	0
$781$	8.92440e6	0.523542
$782$	7.33572e6i	0.428969i
$783$	4.94599e6i	0.288303i
$784$	−614656.	−0.0357143
$785$	0	0
$786$	−4.79522e6	−0.276855
$787$	− 3.06553e7i	− 1.76428i	−0.470984	−	0.882142i	$-0.656101\pi$
$787$	− 3.06553e7i	− 1.76428i	0.470984	−	0.882142i	$-0.343899\pi$
$788$	− 6.54739e6i	− 0.375624i
$789$	2.36525e7	1.35265
$790$	0	0
$791$	1.00721e7	0.572375
$792$	1.01587e7i	0.575474i
$793$	8.70299e6i	0.491457i
$794$	−1.22361e7	−0.688800
$795$	0	0
$796$	4.81568e6	0.269386
$797$	1.51870e7i	0.846886i	0.905923	+	0.423443i	$0.139179\pi$
$797$	1.51870e7i	0.846886i	−0.905923	+	0.423443i	$0.860821\pi$
$798$	− 9.18730e6i	− 0.510718i
$799$	−3.45360e7	−1.91384
$800$	0	0
$801$	−2.04993e7	−1.12891
$802$	1.72317e7i	0.946005i
$803$	− 4.46675e7i	− 2.44457i
$804$	2.43263e7	1.32720
$805$	0	0
$806$	5.49094e6	0.297721
$807$	− 2.86097e7i	− 1.54643i
$808$	3.65760e6i	0.197091i
$809$	−539721.	−0.0289933	−0.0144967	−	0.999895i	$-0.504615\pi$
$809$	−539721.	−0.0289933	−0.0144967	+	0.999895i	$0.504615\pi$
$810$	0	0
$811$	1.39772e7	0.746221	0.373111	−	0.927787i	$-0.378291\pi$
$811$	1.39772e7	0.746221	0.373111	+	0.927787i	$0.378291\pi$
$812$	3.92078e6i	0.208681i
$813$	− 1.69654e7i	− 0.900195i
$814$	−1.24364e7	−0.657862
$815$	0	0
$816$	8.77901e6	0.461551
$817$	1.22688e6i	0.0643051i
$818$	956824.i	0.0499976i
$819$	3.37737e6	0.175942
$820$	0	0
$821$	−1.78137e7	−0.922350	−0.461175	−	0.887309i	$-0.652572\pi$
$821$	−1.78137e7	−0.922350	−0.461175	+	0.887309i	$0.652572\pi$
$822$	1.24631e7i	0.643347i
$823$	1.91010e7i	0.983005i	0.870876	+	0.491502i	$0.163552\pi$
$823$	1.91010e7i	0.983005i	−0.870876	+	0.491502i	$0.836448\pi$
$824$	7.41690e6	0.380543
$825$	0	0
$826$	−1.12190e6	−0.0572144
$827$	− 3.19225e6i	− 0.162305i	−0.996702	−	0.0811526i	$-0.974140\pi$
$827$	− 3.19225e6i	− 0.162305i	0.996702	−	0.0811526i	$-0.0258601\pi$
$828$	− 5.62848e6i	− 0.285309i
$829$	−8.56842e6	−0.433026	−0.216513	−	0.976280i	$-0.569468\pi$
$829$	−8.56842e6	−0.433026	−0.216513	+	0.976280i	$0.569468\pi$
$830$	0	0
$831$	5.06145e7	2.54257
$832$	987136.i	0.0494389i
$833$	− 3.57989e6i	− 0.178755i
$834$	3.21527e7	1.60067
$835$	0	0
$836$	−1.80974e7	−0.895574
$837$	5.63334e6i	0.277941i
$838$	− 1.85385e7i	− 0.911935i
$839$	−3.56751e7	−1.74969	−0.874843	−	0.484407i	$-0.839036\pi$
$839$	−3.56751e7	−1.74969	−0.874843	+	0.484407i	$0.839036\pi$
$840$	0	0
$841$	4.49885e6	0.219337
$842$	− 8.40430e6i	− 0.408528i
$843$	− 4.00152e6i	− 0.193935i
$844$	1.96189e7	0.948021
$845$	0	0
$846$	2.64985e7	1.27290
$847$	− 7.20173e6i	− 0.344928i
$848$	− 6.47578e6i	− 0.309245i
$849$	−1.26742e7	−0.603465
$850$	0	0
$851$	6.89046e6	0.326155
$852$	5.91744e6i	0.279277i
$853$	3.06355e7i	1.44163i	0.693129	+	0.720814i	$0.256232\pi$
$853$	3.06355e7i	1.44163i	−0.693129	+	0.720814i	$0.743768\pi$
$854$	−7.07795e6	−0.332095
$855$	0	0
$856$	8.82317e6	0.411567
$857$	4.46188e6i	0.207523i	0.994602	+	0.103761i	$0.0330879\pi$
$857$	4.46188e6i	0.207523i	−0.994602	+	0.103761i	$0.966912\pi$
$858$	− 1.23055e7i	− 0.570663i
$859$	−2.63974e7	−1.22061	−0.610307	−	0.792165i	$-0.708954\pi$
$859$	−2.63974e7	−1.22061	−0.610307	+	0.792165i	$0.708954\pi$
$860$	0	0
$861$	2.73185e6	0.125588
$862$	6.61936e6i	0.303422i
$863$	− 2.17530e7i	− 0.994244i	−0.867681	−	0.497122i	$-0.834390\pi$
$863$	− 2.17530e7i	− 0.994244i	0.867681	−	0.497122i	$-0.165610\pi$
$864$	−1.01274e6	−0.0461543
$865$	0	0
$866$	7.36122e6	0.333546
$867$	1.84742e7i	0.834674i
$868$	4.46566e6i	0.201181i
$869$	3.56016e7	1.59926
$870$	0	0
$871$	−1.59311e7	−0.711540
$872$	5.65741e6i	0.251957i
$873$	4.31932e7i	1.91814i
$874$	1.00270e7	0.444008
$875$	0	0
$876$	2.96174e7	1.30403
$877$	1.58383e7i	0.695361i	0.937613	+	0.347680i	$0.113031\pi$
$877$	1.58383e7i	0.695361i	−0.937613	+	0.347680i	$0.886969\pi$
$878$	− 2.33474e7i	− 1.02212i
$879$	−3.85277e7	−1.68190
$880$	0	0
$881$	1.97427e7	0.856974	0.428487	−	0.903548i	$-0.359047\pi$
$881$	1.97427e7	0.856974	0.428487	+	0.903548i	$0.359047\pi$
$882$	2.74674e6i	0.118890i
$883$	− 1.72899e7i	− 0.746263i	−0.927779	−	0.373131i	$-0.878284\pi$
$883$	− 1.72899e7i	− 0.746263i	0.927779	−	0.373131i	$-0.121716\pi$
$884$	−5.74930e6	−0.247448
$885$	0	0
$886$	−4.78817e6	−0.204920
$887$	4.44693e7i	1.89780i	0.315574	+	0.948901i	$0.397803\pi$
$887$	4.44693e7i	1.89780i	−0.315574	+	0.948901i	$0.602197\pi$
$888$	− 8.24614e6i	− 0.350928i
$889$	−1.22949e7	−0.521759
$890$	0	0
$891$	−2.59468e7	−1.09494
$892$	− 9.94011e6i	− 0.418292i
$893$	4.72062e7i	1.98094i
$894$	−1.61995e7	−0.677890
$895$	0	0
$896$	−802816.	−0.0334077
$897$	6.81789e6i	0.282923i
$898$	1.37028e7i	0.567046i
$899$	2.84857e7	1.17551
$900$	0	0
$901$	3.77163e7	1.54781
$902$	− 5.38128e6i	− 0.220226i
$903$	− 678454.i	− 0.0276886i
$904$	1.31555e7	0.535408
$905$	0	0
$906$	3.52666e7	1.42739
$907$	− 2.56887e6i	− 0.103687i	−0.998655	−	0.0518434i	$-0.983490\pi$
$907$	− 2.56887e6i	− 0.103687i	0.998655	−	0.0518434i	$-0.0165097\pi$
$908$	2.07628e7i	0.835742i
$909$	1.63449e7	0.656104
$910$	0	0
$911$	1.12692e7	0.449882	0.224941	−	0.974372i	$-0.427781\pi$
$911$	1.12692e7	0.449882	0.224941	+	0.974372i	$0.427781\pi$
$912$	− 1.19997e7i	− 0.477733i
$913$	− 5.89876e7i	− 2.34198i
$914$	2.11897e7	0.838994
$915$	0	0
$916$	−1.98822e6	−0.0782937
$917$	2.55398e6i	0.100298i
$918$	− 5.89840e6i	− 0.231008i
$919$	−3.18378e7	−1.24353	−0.621763	−	0.783205i	$-0.713583\pi$
$919$	−3.18378e7	−1.24353	−0.621763	+	0.783205i	$0.713583\pi$
$920$	0	0
$921$	−5.36039e7	−2.08232
$922$	3.55092e7i	1.37567i
$923$	− 3.87528e6i	− 0.149727i
$924$	1.00078e7	0.385618
$925$	0	0
$926$	8.69901e6	0.333382
$927$	− 3.31443e7i	− 1.26680i
$928$	5.12102e6i	0.195203i
$929$	−1.88558e7	−0.716813	−0.358407	−	0.933566i	$-0.616680\pi$
$929$	−1.88558e7	−0.716813	−0.358407	+	0.933566i	$0.616680\pi$
$930$	0	0
$931$	−4.89324e6	−0.185021
$932$	− 1.73987e7i	− 0.656109i
$933$	− 1.62441e7i	− 0.610931i
$934$	−1.51588e6	−0.0568586
$935$	0	0
$936$	4.41126e6	0.164579
$937$	− 1.12946e7i	− 0.420265i	−0.977673	−	0.210132i	$-0.932610\pi$
$937$	− 1.12946e7i	− 0.420265i	0.977673	−	0.210132i	$-0.0673895\pi$
$938$	− 1.29564e7i	− 0.480814i
$939$	−4.22200e6	−0.156262
$940$	0	0
$941$	−2.91941e7	−1.07478	−0.537392	−	0.843333i	$-0.680590\pi$
$941$	−2.91941e7	−1.07478	−0.537392	+	0.843333i	$0.680590\pi$
$942$	− 3.18241e7i	− 1.16850i
$943$	2.98152e6i	0.109184i
$944$	−1.46534e6	−0.0535192
$945$	0	0
$946$	−1.33644e6	−0.0485536
$947$	− 1.05892e7i	− 0.383697i	−0.981425	−	0.191848i	$-0.938552\pi$
$947$	− 1.05892e7i	− 0.383697i	0.981425	−	0.191848i	$-0.0614482\pi$
$948$	2.36061e7i	0.853107i
$949$	−1.93962e7	−0.699118
$950$	0	0
$951$	−6.22530e7	−2.23208
$952$	− 4.67578e6i	− 0.167210i
$953$	− 3.90317e7i	− 1.39215i	−0.717970	−	0.696074i	$-0.754929\pi$
$953$	− 3.90317e7i	− 1.39215i	0.717970	−	0.696074i	$-0.245071\pi$
$954$	−2.89386e7	−1.02945
$955$	0	0
$956$	−8.73010e6	−0.308940
$957$	− 6.38378e7i	− 2.25319i
$958$	− 7.53958e6i	− 0.265420i
$959$	6.63793e6	0.233070
$960$	0	0
$961$	3.81526e6	0.133265
$962$	5.40033e6i	0.188141i
$963$	− 3.94285e7i	− 1.37008i
$964$	−1.29810e7	−0.449898
$965$	0	0
$966$	−5.54484e6	−0.191182
$967$	3.43395e7i	1.18094i	0.807059	+	0.590470i	$0.201058\pi$
$967$	3.43395e7i	1.18094i	−0.807059	+	0.590470i	$0.798942\pi$
$968$	− 9.40634e6i	− 0.322650i
$969$	6.98891e7	2.39111
$970$	0	0
$971$	−1.81464e7	−0.617651	−0.308826	−	0.951119i	$-0.599936\pi$
$971$	−1.81464e7	−0.617651	−0.308826	+	0.951119i	$0.599936\pi$
$972$	− 2.10496e7i	− 0.714625i
$973$	− 1.71248e7i	− 0.579888i
$974$	1.47075e7	0.496756
$975$	0	0
$976$	−9.24467e6	−0.310647
$977$	1.05223e7i	0.352675i	0.984330	+	0.176338i	$0.0564250\pi$
$977$	1.05223e7i	0.352675i	−0.984330	+	0.176338i	$0.943575\pi$
$978$	8.42591e6i	0.281689i
$979$	3.97802e7	1.32651
$980$	0	0
$981$	2.52815e7	0.838747
$982$	3.81606e7i	1.26281i
$983$	− 7.91353e6i	− 0.261208i	−0.991435	−	0.130604i	$-0.958308\pi$
$983$	− 7.91353e6i	− 0.261208i	0.991435	−	0.130604i	$-0.0416916\pi$
$984$	3.56813e6	0.117477
$985$	0	0
$986$	−2.98260e7	−0.977017
$987$	− 2.61047e7i	− 0.852954i
$988$	7.85853e6i	0.256123i
$989$	740460.	0.0240719
$990$	0	0
$991$	4.01556e7	1.29886	0.649429	−	0.760422i	$-0.275008\pi$
$991$	4.01556e7	1.29886	0.649429	+	0.760422i	$0.275008\pi$
$992$	5.83270e6i	0.188187i
$993$	4.92976e7i	1.58654i
$994$	3.15168e6	0.101176
$995$	0	0
$996$	3.91125e7	1.24930
$997$	4.93478e7i	1.57228i	0.618048	+	0.786140i	$0.287924\pi$
$997$	4.93478e7i	1.57228i	−0.618048	+	0.786140i	$0.712076\pi$
$998$	− 1.91297e7i	− 0.607970i
$999$	−5.54038e6	−0.175641

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.c.h.99.2		2
5.2	odd	4		70.6.a.a.1.1	✓	1
5.3	odd	4		350.6.a.n.1.1		1
5.4	even	2	inner	350.6.c.h.99.1		2
15.2	even	4		630.6.a.j.1.1		1
20.7	even	4		560.6.a.i.1.1		1
35.27	even	4		490.6.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.6.a.a.1.1	✓	1	5.2	odd	4
350.6.a.n.1.1		1	5.3	odd	4
350.6.c.h.99.1		2	5.4	even	2	inner
350.6.c.h.99.2		2	1.1	even	1	trivial
490.6.a.i.1.1		1	35.27	even	4
560.6.a.i.1.1		1	20.7	even	4
630.6.a.j.1.1		1	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -23.0000i q^{3} -16.0000 q^{4} +92.0000 q^{6} -49.0000i q^{7} -64.0000i q^{8} -286.000 q^{9} +O(q^{10})\)	\(q+4.00000i q^{2} -23.0000i q^{3} -16.0000 q^{4} +92.0000 q^{6} -49.0000i q^{7} -64.0000i q^{8} -286.000 q^{9} +555.000 q^{11} +368.000i q^{12} -241.000i q^{13} +196.000 q^{14} +256.000 q^{16} +1491.00i q^{17} -1144.00i q^{18} +2038.00 q^{19} -1127.00 q^{21} +2220.00i q^{22} -1230.00i q^{23} -1472.00 q^{24} +964.000 q^{26} +989.000i q^{27} +784.000i q^{28} +5001.00 q^{29} +5696.00 q^{31} +1024.00i q^{32} -12765.0i q^{33} -5964.00 q^{34} +4576.00 q^{36} +5602.00i q^{37} +8152.00i q^{38} -5543.00 q^{39} -2424.00 q^{41} -4508.00i q^{42} +602.000i q^{43} -8880.00 q^{44} +4920.00 q^{46} +23163.0i q^{47} -5888.00i q^{48} -2401.00 q^{49} +34293.0 q^{51} +3856.00i q^{52} -25296.0i q^{53} -3956.00 q^{54} -3136.00 q^{56} -46874.0i q^{57} +20004.0i q^{58} -5724.00 q^{59} -36112.0 q^{61} +22784.0i q^{62} +14014.0i q^{63} -4096.00 q^{64} +51060.0 q^{66} -66104.0i q^{67} -23856.0i q^{68} -28290.0 q^{69} +16080.0 q^{71} +18304.0i q^{72} -80482.0i q^{73} -22408.0 q^{74} -32608.0 q^{76} -27195.0i q^{77} -22172.0i q^{78} +64147.0 q^{79} -46751.0 q^{81} -9696.00i q^{82} -106284. i q^{83} +18032.0 q^{84} -2408.00 q^{86} -115023. i q^{87} -35520.0i q^{88} +71676.0 q^{89} -11809.0 q^{91} +19680.0i q^{92} -131008. i q^{93} -92652.0 q^{94} +23552.0 q^{96} -151025. i q^{97} -9604.00i q^{98} -158730. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} + 184 q^{6} - 572 q^{9}+O(q^{10}) \) 2 * q - 32 * q^4 + 184 * q^6 - 572 * q^9	\( 2 q - 32 q^{4} + 184 q^{6} - 572 q^{9} + 1110 q^{11} + 392 q^{14} + 512 q^{16} + 4076 q^{19} - 2254 q^{21} - 2944 q^{24} + 1928 q^{26} + 10002 q^{29} + 11392 q^{31} - 11928 q^{34} + 9152 q^{36} - 11086 q^{39} - 4848 q^{41} - 17760 q^{44} + 9840 q^{46} - 4802 q^{49} + 68586 q^{51} - 7912 q^{54} - 6272 q^{56} - 11448 q^{59} - 72224 q^{61} - 8192 q^{64} + 102120 q^{66} - 56580 q^{69} + 32160 q^{71} - 44816 q^{74} - 65216 q^{76} + 128294 q^{79} - 93502 q^{81} + 36064 q^{84} - 4816 q^{86} + 143352 q^{89} - 23618 q^{91} - 185304 q^{94} + 47104 q^{96} - 317460 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 + 184 * q^6 - 572 * q^9 + 1110 * q^11 + 392 * q^14 + 512 * q^16 + 4076 * q^19 - 2254 * q^21 - 2944 * q^24 + 1928 * q^26 + 10002 * q^29 + 11392 * q^31 - 11928 * q^34 + 9152 * q^36 - 11086 * q^39 - 4848 * q^41 - 17760 * q^44 + 9840 * q^46 - 4802 * q^49 + 68586 * q^51 - 7912 * q^54 - 6272 * q^56 - 11448 * q^59 - 72224 * q^61 - 8192 * q^64 + 102120 * q^66 - 56580 * q^69 + 32160 * q^71 - 44816 * q^74 - 65216 * q^76 + 128294 * q^79 - 93502 * q^81 + 36064 * q^84 - 4816 * q^86 + 143352 * q^89 - 23618 * q^91 - 185304 * q^94 + 47104 * q^96 - 317460 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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