LMFDB - Embedded newform 676.4.d.d.337.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [676,4,Mod(337,676)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(676, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("676.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 676 = 2^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	676.d (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:	$39.8852911639$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 51x^{6} - 224x^{5} + 2520x^{4} - 5712x^{3} + 16675x^{2} + 9072x + 6561 $ x^8 + 51x^6 - 224x^5 + 2520x^4 - 5712x^3 + 16675x^2 + 9072x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3 $
Twist minimal:	no (minimal twist has level 52)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.8
Root			$1.73860 + 3.01134i$ of defining polynomial
Character	$\chi$	$=$	676.337
Dual form			676.4.d.d.337.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.96207 q^{3} +11.8097i q^{5} +32.1873i q^{7} +53.3187 q^{9} +O(q^{10})$	$q+8.96207 q^{3} +11.8097i q^{5} +32.1873i q^{7} +53.3187 q^{9} -12.0853i q^{11} +105.839i q^{15} -89.7581 q^{17} +19.1714i q^{19} +288.465i q^{21} -15.7774 q^{23} -14.4680 q^{25} +235.870 q^{27} +262.394 q^{29} +84.0297i q^{31} -108.309i q^{33} -380.121 q^{35} -271.198i q^{37} +165.224i q^{41} -270.307 q^{43} +629.676i q^{45} +238.811i q^{47} -693.021 q^{49} -804.419 q^{51} -94.2765 q^{53} +142.723 q^{55} +171.816i q^{57} +247.310i q^{59} +203.694 q^{61} +1716.18i q^{63} -231.451i q^{67} -141.398 q^{69} -749.727i q^{71} -153.347i q^{73} -129.663 q^{75} +388.991 q^{77} -881.413 q^{79} +674.281 q^{81} -197.016i q^{83} -1060.01i q^{85} +2351.60 q^{87} +1390.64i q^{89} +753.080i q^{93} -226.408 q^{95} +1122.87i q^{97} -644.370i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 140 q^{9}+O(q^{10}) $ 8 * q + 140 * q^9	$ 8 q + 140 q^{9} - 176 q^{17} + 40 q^{23} - 84 q^{25} - 432 q^{27} + 968 q^{29} - 80 q^{35} - 1008 q^{43} - 1844 q^{49} - 1808 q^{51} - 1164 q^{53} + 2256 q^{55} + 2448 q^{61} - 3476 q^{69} - 2896 q^{75} - 3972 q^{77} - 3968 q^{79} + 8264 q^{81} + 3320 q^{87} - 4800 q^{95}+O(q^{100}) $ 8 * q + 140 * q^9 - 176 * q^17 + 40 * q^23 - 84 * q^25 - 432 * q^27 + 968 * q^29 - 80 * q^35 - 1008 * q^43 - 1844 * q^49 - 1808 * q^51 - 1164 * q^53 + 2256 * q^55 + 2448 * q^61 - 3476 * q^69 - 2896 * q^75 - 3972 * q^77 - 3968 * q^79 + 8264 * q^81 + 3320 * q^87 - 4800 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/676\mathbb{Z}\right)^\times$.

$n$	$339$	$509$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	8.96207	1.72475	0.862376	−	0.506269i	$-0.168976\pi$
$3$	8.96207	1.72475	0.862376	+	0.506269i	$0.168976\pi$
$4$	0	0
$5$	11.8097i	1.05629i	0.849155	+	0.528144i	$0.177112\pi$
$5$	11.8097i	1.05629i	−0.849155	+	0.528144i	$0.822888\pi$
$6$	0	0
$7$	32.1873i	1.73795i	0.494856	+	0.868975i	$0.335221\pi$
$7$	32.1873i	1.73795i	−0.494856	+	0.868975i	$0.664779\pi$
$8$	0	0
$9$	53.3187	1.97477
$10$	0	0
$11$	− 12.0853i	− 0.331258i	−0.986188	−	0.165629i	$-0.947035\pi$
$11$	− 12.0853i	− 0.331258i	0.986188	−	0.165629i	$-0.0529655\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	105.839i	1.82183i
$16$	0	0
$17$	−89.7581	−1.28056	−0.640281	−	0.768141i	$-0.721182\pi$
$17$	−89.7581	−1.28056	−0.640281	+	0.768141i	$0.721182\pi$
$18$	0	0
$19$	19.1714i	0.231486i	0.993279	+	0.115743i	$0.0369249\pi$
$19$	19.1714i	0.231486i	−0.993279	+	0.115743i	$0.963075\pi$
$20$	0	0
$21$	288.465i	2.99753i
$22$	0	0
$23$	−15.7774	−0.143035	−0.0715176	−	0.997439i	$-0.522784\pi$
$23$	−15.7774	−0.143035	−0.0715176	+	0.997439i	$0.522784\pi$
$24$	0	0
$25$	−14.4680	−0.115744
$26$	0	0
$27$	235.870	1.68123
$28$	0	0
$29$	262.394	1.68019	0.840093	−	0.542442i	$-0.182500\pi$
$29$	262.394	1.68019	0.840093	+	0.542442i	$0.182500\pi$
$30$	0	0
$31$	84.0297i	0.486844i	0.969920	+	0.243422i	$0.0782700\pi$
$31$	84.0297i	0.486844i	−0.969920	+	0.243422i	$0.921730\pi$
$32$	0	0
$33$	− 108.309i	− 0.571338i
$34$	0	0
$35$	−380.121	−1.83578
$36$	0	0
$37$	− 271.198i	− 1.20499i	−0.798123	−	0.602495i	$-0.794173\pi$
$37$	− 271.198i	− 1.20499i	0.798123	−	0.602495i	$-0.205827\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	165.224i	0.629359i	0.949198	+	0.314679i	$0.101897\pi$
$41$	165.224i	0.629359i	−0.949198	+	0.314679i	$0.898103\pi$
$42$	0	0
$43$	−270.307	−0.958637	−0.479318	−	0.877641i	$-0.659116\pi$
$43$	−270.307	−0.958637	−0.479318	+	0.877641i	$0.659116\pi$
$44$	0	0
$45$	629.676i	2.08592i
$46$	0	0
$47$	238.811i	0.741153i	0.928802	+	0.370577i	$0.120840\pi$
$47$	238.811i	0.741153i	−0.928802	+	0.370577i	$0.879160\pi$
$48$	0	0
$49$	−693.021	−2.02047
$50$	0	0
$51$	−804.419	−2.20865
$52$	0	0
$53$	−94.2765	−0.244337	−0.122169	−	0.992509i	$-0.538985\pi$
$53$	−94.2765	−0.244337	−0.122169	+	0.992509i	$0.538985\pi$
$54$	0	0
$55$	142.723	0.349904
$56$	0	0
$57$	171.816i	0.399255i
$58$	0	0
$59$	247.310i	0.545713i	0.962055	+	0.272856i	$0.0879684\pi$
$59$	247.310i	0.545713i	−0.962055	+	0.272856i	$0.912032\pi$
$60$	0	0
$61$	203.694	0.427547	0.213774	−	0.976883i	$-0.431424\pi$
$61$	203.694	0.427547	0.213774	+	0.976883i	$0.431424\pi$
$62$	0	0
$63$	1716.18i	3.43205i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 231.451i	− 0.422033i	−0.977483	−	0.211016i	$-0.932323\pi$
$67$	− 231.451i	− 0.422033i	0.977483	−	0.211016i	$-0.0676774\pi$
$68$	0	0
$69$	−141.398	−0.246700
$70$	0	0
$71$	− 749.727i	− 1.25319i	−0.779347	−	0.626593i	$-0.784449\pi$
$71$	− 749.727i	− 1.25319i	0.779347	−	0.626593i	$-0.215551\pi$
$72$	0	0
$73$	− 153.347i	− 0.245862i	−0.992415	−	0.122931i	$-0.960771\pi$
$73$	− 153.347i	− 0.245862i	0.992415	−	0.122931i	$-0.0392294\pi$
$74$	0	0
$75$	−129.663	−0.199630
$76$	0	0
$77$	388.991	0.575710
$78$	0	0
$79$	−881.413	−1.25527	−0.627637	−	0.778506i	$-0.715978\pi$
$79$	−881.413	−1.25527	−0.627637	+	0.778506i	$0.715978\pi$
$80$	0	0
$81$	674.281	0.924939
$82$	0	0
$83$	− 197.016i	− 0.260546i	−0.991478	−	0.130273i	$-0.958415\pi$
$83$	− 197.016i	− 0.260546i	0.991478	−	0.130273i	$-0.0415854\pi$
$84$	0	0
$85$	− 1060.01i	− 1.35264i
$86$	0	0
$87$	2351.60	2.89790
$88$	0	0
$89$	1390.64i	1.65627i	0.560532	+	0.828133i	$0.310597\pi$
$89$	1390.64i	1.65627i	−0.560532	+	0.828133i	$0.689403\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	753.080i	0.839685i
$94$	0	0
$95$	−226.408	−0.244516
$96$	0	0
$97$	1122.87i	1.17536i	0.809094	+	0.587679i	$0.199958\pi$
$97$	1122.87i	1.17536i	−0.809094	+	0.587679i	$0.800042\pi$
$98$	0	0
$99$	− 644.370i	− 0.654158i
$100$	0	0
$101$	1193.40	1.17572	0.587862	−	0.808961i	$-0.299970\pi$
$101$	1193.40	1.17572	0.587862	+	0.808961i	$0.299970\pi$
$102$	0	0
$103$	1367.61	1.30830	0.654149	−	0.756366i	$-0.273027\pi$
$103$	1367.61	1.30830	0.654149	+	0.756366i	$0.273027\pi$
$104$	0	0
$105$	−3406.67	−3.16626
$106$	0	0
$107$	1736.87	1.56925	0.784623	−	0.619973i	$-0.212856\pi$
$107$	1736.87	1.56925	0.784623	+	0.619973i	$0.212856\pi$
$108$	0	0
$109$	− 946.465i	− 0.831697i	−0.909434	−	0.415848i	$-0.863485\pi$
$109$	− 946.465i	− 0.831697i	0.909434	−	0.415848i	$-0.136515\pi$
$110$	0	0
$111$	− 2430.49i	− 2.07831i
$112$	0	0
$113$	−971.425	−0.808708	−0.404354	−	0.914603i	$-0.632504\pi$
$113$	−971.425	−0.808708	−0.404354	+	0.914603i	$0.632504\pi$
$114$	0	0
$115$	− 186.325i	− 0.151086i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 2889.07i	− 2.22555i
$120$	0	0
$121$	1184.95	0.890268
$122$	0	0
$123$	1480.75i	1.08549i
$124$	0	0
$125$	1305.35i	0.934029i
$126$	0	0
$127$	1603.86	1.12063	0.560313	−	0.828281i	$-0.310681\pi$
$127$	1603.86	1.12063	0.560313	+	0.828281i	$0.310681\pi$
$128$	0	0
$129$	−2422.51	−1.65341
$130$	0	0
$131$	1677.20	1.11861	0.559305	−	0.828962i	$-0.311068\pi$
$131$	1677.20	1.11861	0.559305	+	0.828962i	$0.311068\pi$
$132$	0	0
$133$	−617.076	−0.402311
$134$	0	0
$135$	2785.55i	1.77586i
$136$	0	0
$137$	− 584.065i	− 0.364234i	−0.983277	−	0.182117i	$-0.941705\pi$
$137$	− 584.065i	− 0.364234i	0.983277	−	0.182117i	$-0.0582949\pi$
$138$	0	0
$139$	156.610	0.0955646	0.0477823	−	0.998858i	$-0.484785\pi$
$139$	156.610	0.0955646	0.0477823	+	0.998858i	$0.484785\pi$
$140$	0	0
$141$	2140.24i	1.27831i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	3098.79i	1.77476i
$146$	0	0
$147$	−6210.91	−3.48481
$148$	0	0
$149$	1187.02i	0.652645i	0.945259	+	0.326322i	$0.105809\pi$
$149$	1187.02i	0.652645i	−0.945259	+	0.326322i	$0.894191\pi$
$150$	0	0
$151$	− 2022.29i	− 1.08988i	−0.838476	−	0.544939i	$-0.816553\pi$
$151$	− 2022.29i	− 1.08988i	0.838476	−	0.544939i	$-0.183447\pi$
$152$	0	0
$153$	−4785.79	−2.52881
$154$	0	0
$155$	−992.361	−0.514248
$156$	0	0
$157$	1429.86	0.726848	0.363424	−	0.931624i	$-0.381608\pi$
$157$	1429.86	0.726848	0.363424	+	0.931624i	$0.381608\pi$
$158$	0	0
$159$	−844.913	−0.421421
$160$	0	0
$161$	− 507.831i	− 0.248588i
$162$	0	0
$163$	1110.69i	0.533719i	0.963735	+	0.266860i	$0.0859860\pi$
$163$	1110.69i	0.533719i	−0.963735	+	0.266860i	$0.914014\pi$
$164$	0	0
$165$	1279.09	0.603497
$166$	0	0
$167$	− 284.371i	− 0.131768i	−0.997827	−	0.0658841i	$-0.979013\pi$
$167$	− 284.371i	− 0.131768i	0.997827	−	0.0658841i	$-0.0209867\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	1022.20i	0.457130i
$172$	0	0
$173$	4340.82	1.90767	0.953834	−	0.300334i	$-0.0970981\pi$
$173$	4340.82	1.90767	0.953834	+	0.300334i	$0.0970981\pi$
$174$	0	0
$175$	− 465.686i	− 0.201157i
$176$	0	0
$177$	2216.41i	0.941219i
$178$	0	0
$179$	−650.789	−0.271744	−0.135872	−	0.990726i	$-0.543384\pi$
$179$	−650.789	−0.271744	−0.135872	+	0.990726i	$0.543384\pi$
$180$	0	0
$181$	454.138	0.186496	0.0932482	−	0.995643i	$-0.470275\pi$
$181$	454.138	0.186496	0.0932482	+	0.995643i	$0.470275\pi$
$182$	0	0
$183$	1825.52	0.737413
$184$	0	0
$185$	3202.75	1.27282
$186$	0	0
$187$	1084.75i	0.424197i
$188$	0	0
$189$	7592.02i	2.92190i
$190$	0	0
$191$	−2815.26	−1.06652	−0.533259	−	0.845952i	$-0.679033\pi$
$191$	−2815.26	−1.06652	−0.533259	+	0.845952i	$0.679033\pi$
$192$	0	0
$193$	− 1153.57i	− 0.430238i	−0.976588	−	0.215119i	$-0.930986\pi$
$193$	− 1153.57i	− 0.430238i	0.976588	−	0.215119i	$-0.0690140\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 406.657i	− 0.147072i	−0.997293	−	0.0735359i	$-0.976572\pi$
$197$	− 406.657i	− 0.147072i	0.997293	−	0.0735359i	$-0.0234283\pi$
$198$	0	0
$199$	809.398	0.288325	0.144163	−	0.989554i	$-0.453951\pi$
$199$	809.398	0.288325	0.144163	+	0.989554i	$0.453951\pi$
$200$	0	0
$201$	− 2074.28i	− 0.727902i
$202$	0	0
$203$	8445.76i	2.92008i
$204$	0	0
$205$	−1951.24	−0.664784
$206$	0	0
$207$	−841.229	−0.282461
$208$	0	0
$209$	231.692	0.0766815
$210$	0	0
$211$	3600.54	1.17475	0.587373	−	0.809317i	$-0.300162\pi$
$211$	3600.54	1.17475	0.587373	+	0.809317i	$0.300162\pi$
$212$	0	0
$213$	− 6719.10i	− 2.16143i
$214$	0	0
$215$	− 3192.23i	− 1.01260i
$216$	0	0
$217$	−2704.69	−0.846111
$218$	0	0
$219$	− 1374.31i	− 0.424051i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	2610.80i	0.784000i	0.919965	+	0.392000i	$0.128217\pi$
$223$	2610.80i	0.784000i	−0.919965	+	0.392000i	$0.871783\pi$
$224$	0	0
$225$	−771.415	−0.228567
$226$	0	0
$227$	− 4660.87i	− 1.36279i	−0.731917	−	0.681394i	$-0.761374\pi$
$227$	− 4660.87i	− 1.36279i	0.731917	−	0.681394i	$-0.238626\pi$
$228$	0	0
$229$	123.893i	0.0357515i	0.999840	+	0.0178757i	$0.00569032\pi$
$229$	123.893i	0.0357515i	−0.999840	+	0.0178757i	$0.994310\pi$
$230$	0	0
$231$	3486.17	0.992957
$232$	0	0
$233$	−1186.44	−0.333590	−0.166795	−	0.985992i	$-0.553342\pi$
$233$	−1186.44	−0.333590	−0.166795	+	0.985992i	$0.553342\pi$
$234$	0	0
$235$	−2820.28	−0.782871
$236$	0	0
$237$	−7899.29	−2.16504
$238$	0	0
$239$	4543.56i	1.22970i	0.788644	+	0.614850i	$0.210783\pi$
$239$	4543.56i	1.22970i	−0.788644	+	0.614850i	$0.789217\pi$
$240$	0	0
$241$	− 6284.16i	− 1.67966i	−0.542849	−	0.839831i	$-0.682654\pi$
$241$	− 6284.16i	− 1.67966i	0.542849	−	0.839831i	$-0.317346\pi$
$242$	0	0
$243$	−325.546	−0.0859415
$244$	0	0
$245$	− 8184.34i	− 2.13420i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 1765.67i	− 0.449377i
$250$	0	0
$251$	−924.809	−0.232563	−0.116282	−	0.993216i	$-0.537098\pi$
$251$	−924.809	−0.232563	−0.116282	+	0.993216i	$0.537098\pi$
$252$	0	0
$253$	190.674i	0.0473816i
$254$	0	0
$255$	− 9499.91i	− 2.33297i
$256$	0	0
$257$	542.411	0.131652	0.0658261	−	0.997831i	$-0.479032\pi$
$257$	542.411	0.131652	0.0658261	+	0.997831i	$0.479032\pi$
$258$	0	0
$259$	8729.11	2.09421
$260$	0	0
$261$	13990.5	3.31798
$262$	0	0
$263$	−5248.79	−1.23062	−0.615312	−	0.788283i	$-0.710970\pi$
$263$	−5248.79	−1.23062	−0.615312	+	0.788283i	$0.710970\pi$
$264$	0	0
$265$	− 1113.37i	− 0.258091i
$266$	0	0
$267$	12463.0i	2.85665i
$268$	0	0
$269$	−330.160	−0.0748336	−0.0374168	−	0.999300i	$-0.511913\pi$
$269$	−330.160	−0.0748336	−0.0374168	+	0.999300i	$0.511913\pi$
$270$	0	0
$271$	− 1433.88i	− 0.321409i	−0.987003	−	0.160705i	$-0.948623\pi$
$271$	− 1433.88i	− 0.321409i	0.987003	−	0.160705i	$-0.0513767\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	174.849i	0.0383411i
$276$	0	0
$277$	2141.07	0.464421	0.232211	−	0.972666i	$-0.425404\pi$
$277$	2141.07	0.464421	0.232211	+	0.972666i	$0.425404\pi$
$278$	0	0
$279$	4480.35i	0.961404i
$280$	0	0
$281$	− 5857.04i	− 1.24342i	−0.783246	−	0.621711i	$-0.786438\pi$
$281$	− 5857.04i	− 1.24342i	0.783246	−	0.621711i	$-0.213562\pi$
$282$	0	0
$283$	1707.50	0.358657	0.179329	−	0.983789i	$-0.442607\pi$
$283$	1707.50	0.358657	0.179329	+	0.983789i	$0.442607\pi$
$284$	0	0
$285$	−2029.09	−0.421729
$286$	0	0
$287$	−5318.12	−1.09379
$288$	0	0
$289$	3143.52	0.639838
$290$	0	0
$291$	10063.2i	2.02720i
$292$	0	0
$293$	− 1783.29i	− 0.355565i	−0.984070	−	0.177783i	$-0.943108\pi$
$293$	− 1783.29i	− 0.355565i	0.984070	−	0.177783i	$-0.0568924\pi$
$294$	0	0
$295$	−2920.65	−0.576430
$296$	0	0
$297$	− 2850.55i	− 0.556922i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 8700.44i	− 1.66606i
$302$	0	0
$303$	10695.4	2.02783
$304$	0	0
$305$	2405.56i	0.451613i
$306$	0	0
$307$	− 4027.85i	− 0.748801i	−0.927267	−	0.374400i	$-0.877849\pi$
$307$	− 4027.85i	− 0.748801i	0.927267	−	0.374400i	$-0.122151\pi$
$308$	0	0
$309$	12256.6	2.25649
$310$	0	0
$311$	−80.6308	−0.0147014	−0.00735072	−	0.999973i	$-0.502340\pi$
$311$	−80.6308	−0.0147014	−0.00735072	+	0.999973i	$0.502340\pi$
$312$	0	0
$313$	−4628.22	−0.835790	−0.417895	−	0.908495i	$-0.637232\pi$
$313$	−4628.22	−0.835790	−0.417895	+	0.908495i	$0.637232\pi$
$314$	0	0
$315$	−20267.6	−3.62523
$316$	0	0
$317$	− 10723.3i	− 1.89993i	−0.312354	−	0.949966i	$-0.601118\pi$
$317$	− 10723.3i	− 1.89993i	0.312354	−	0.949966i	$-0.398882\pi$
$318$	0	0
$319$	− 3171.10i	− 0.556576i
$320$	0	0
$321$	15565.9	2.70656
$322$	0	0
$323$	− 1720.79i	− 0.296432i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 8482.29i	− 1.43447i
$328$	0	0
$329$	−7686.69	−1.28809
$330$	0	0
$331$	− 5192.52i	− 0.862256i	−0.902291	−	0.431128i	$-0.858116\pi$
$331$	− 5192.52i	− 0.862256i	0.902291	−	0.431128i	$-0.141884\pi$
$332$	0	0
$333$	− 14459.9i	− 2.37957i
$334$	0	0
$335$	2733.35	0.445788
$336$	0	0
$337$	2676.49	0.432635	0.216317	−	0.976323i	$-0.430595\pi$
$337$	2676.49	0.432635	0.216317	+	0.976323i	$0.430595\pi$
$338$	0	0
$339$	−8705.98	−1.39482
$340$	0	0
$341$	1015.52	0.161271
$342$	0	0
$343$	− 11266.2i	− 1.77353i
$344$	0	0
$345$	− 1669.86i	− 0.260586i
$346$	0	0
$347$	−9848.46	−1.52361	−0.761806	−	0.647806i	$-0.775687\pi$
$347$	−9848.46	−1.52361	−0.761806	+	0.647806i	$0.775687\pi$
$348$	0	0
$349$	− 6511.34i	− 0.998693i	−0.866402	−	0.499347i	$-0.833573\pi$
$349$	− 6511.34i	− 0.998693i	0.866402	−	0.499347i	$-0.166427\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11096.5i	1.67310i	0.547888	+	0.836552i	$0.315432\pi$
$353$	11096.5i	1.67310i	−0.547888	+	0.836552i	$0.684568\pi$
$354$	0	0
$355$	8854.01	1.32372
$356$	0	0
$357$	− 25892.1i	− 3.83852i
$358$	0	0
$359$	7844.27i	1.15322i	0.817021	+	0.576608i	$0.195624\pi$
$359$	7844.27i	1.15322i	−0.817021	+	0.576608i	$0.804376\pi$
$360$	0	0
$361$	6491.46	0.946414
$362$	0	0
$363$	10619.6	1.53549
$364$	0	0
$365$	1810.98	0.259701
$366$	0	0
$367$	3275.59	0.465898	0.232949	−	0.972489i	$-0.425162\pi$
$367$	3275.59	0.465898	0.232949	+	0.972489i	$0.425162\pi$
$368$	0	0
$369$	8809.55i	1.24284i
$370$	0	0
$371$	− 3034.51i	− 0.424646i
$372$	0	0
$373$	−1625.30	−0.225616	−0.112808	−	0.993617i	$-0.535985\pi$
$373$	−1625.30	−0.225616	−0.112808	+	0.993617i	$0.535985\pi$
$374$	0	0
$375$	11698.6i	1.61097i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 12919.0i	− 1.75094i	−0.483273	−	0.875470i	$-0.660552\pi$
$379$	− 12919.0i	− 1.75094i	0.483273	−	0.875470i	$-0.339448\pi$
$380$	0	0
$381$	14373.9	1.93280
$382$	0	0
$383$	4965.91i	0.662523i	0.943539	+	0.331261i	$0.107474\pi$
$383$	4965.91i	0.662523i	−0.943539	+	0.331261i	$0.892526\pi$
$384$	0	0
$385$	4593.86i	0.608116i
$386$	0	0
$387$	−14412.4	−1.89309
$388$	0	0
$389$	8704.29	1.13451	0.567256	−	0.823542i	$-0.308005\pi$
$389$	8704.29	1.13451	0.567256	+	0.823542i	$0.308005\pi$
$390$	0	0
$391$	1416.15	0.183165
$392$	0	0
$393$	15031.2	1.92932
$394$	0	0
$395$	− 10409.2i	− 1.32593i
$396$	0	0
$397$	4394.23i	0.555517i	0.960651	+	0.277759i	$0.0895915\pi$
$397$	4394.23i	0.555517i	−0.960651	+	0.277759i	$0.910408\pi$
$398$	0	0
$399$	−5530.28	−0.693886
$400$	0	0
$401$	− 3249.70i	− 0.404694i	−0.979314	−	0.202347i	$-0.935143\pi$
$401$	− 3249.70i	− 0.404694i	0.979314	−	0.202347i	$-0.0648568\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	7963.02i	0.977002i
$406$	0	0
$407$	−3277.49	−0.399163
$408$	0	0
$409$	− 6565.42i	− 0.793739i	−0.917875	−	0.396870i	$-0.870097\pi$
$409$	− 6565.42i	− 0.793739i	0.917875	−	0.396870i	$-0.129903\pi$
$410$	0	0
$411$	− 5234.43i	− 0.628213i
$412$	0	0
$413$	−7960.25	−0.948422
$414$	0	0
$415$	2326.69	0.275212
$416$	0	0
$417$	1403.55	0.164825
$418$	0	0
$419$	−11364.7	−1.32506	−0.662532	−	0.749033i	$-0.730518\pi$
$419$	−11364.7	−1.32506	−0.662532	+	0.749033i	$0.730518\pi$
$420$	0	0
$421$	3657.27i	0.423383i	0.977337	+	0.211692i	$0.0678972\pi$
$421$	3657.27i	0.423383i	−0.977337	+	0.211692i	$0.932103\pi$
$422$	0	0
$423$	12733.1i	1.46361i
$424$	0	0
$425$	1298.62	0.148217
$426$	0	0
$427$	6556.37i	0.743056i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13367.7i	1.49396i	0.664844	+	0.746982i	$0.268498\pi$
$431$	13367.7i	1.49396i	−0.664844	+	0.746982i	$0.731502\pi$
$432$	0	0
$433$	−6062.75	−0.672880	−0.336440	−	0.941705i	$-0.609223\pi$
$433$	−6062.75	−0.672880	−0.336440	+	0.941705i	$0.609223\pi$
$434$	0	0
$435$	27771.6i	3.06102i
$436$	0	0
$437$	− 302.475i	− 0.0331106i
$438$	0	0
$439$	1711.42	0.186063	0.0930315	−	0.995663i	$-0.470344\pi$
$439$	1711.42	0.186063	0.0930315	+	0.995663i	$0.470344\pi$
$440$	0	0
$441$	−36951.0	−3.98996
$442$	0	0
$443$	14253.1	1.52864	0.764320	−	0.644837i	$-0.223075\pi$
$443$	14253.1	1.52864	0.764320	+	0.644837i	$0.223075\pi$
$444$	0	0
$445$	−16423.0	−1.74949
$446$	0	0
$447$	10638.1i	1.12565i
$448$	0	0
$449$	3532.49i	0.371289i	0.982617	+	0.185644i	$0.0594372\pi$
$449$	3532.49i	0.371289i	−0.982617	+	0.185644i	$0.940563\pi$
$450$	0	0
$451$	1996.78	0.208480
$452$	0	0
$453$	− 18123.9i	− 1.87977i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 5587.49i	− 0.571929i	−0.958240	−	0.285965i	$-0.907686\pi$
$457$	− 5587.49i	− 0.571929i	0.958240	−	0.285965i	$-0.0923140\pi$
$458$	0	0
$459$	−21171.3	−2.15292
$460$	0	0
$461$	− 2688.18i	− 0.271586i	−0.990737	−	0.135793i	$-0.956642\pi$
$461$	− 2688.18i	− 0.271586i	0.990737	−	0.135793i	$-0.0433582\pi$
$462$	0	0
$463$	19314.2i	1.93868i	0.245723	+	0.969340i	$0.420975\pi$
$463$	19314.2i	1.93868i	−0.245723	+	0.969340i	$0.579025\pi$
$464$	0	0
$465$	−8893.61	−0.886949
$466$	0	0
$467$	−5482.16	−0.543221	−0.271611	−	0.962407i	$-0.587556\pi$
$467$	−5482.16	−0.543221	−0.271611	+	0.962407i	$0.587556\pi$
$468$	0	0
$469$	7449.77	0.733472
$470$	0	0
$471$	12814.5	1.25363
$472$	0	0
$473$	3266.72i	0.317556i
$474$	0	0
$475$	− 277.372i	− 0.0267931i
$476$	0	0
$477$	−5026.71	−0.482509
$478$	0	0
$479$	− 5068.54i	− 0.483481i	−0.970341	−	0.241740i	$-0.922282\pi$
$479$	− 5068.54i	− 0.483481i	0.970341	−	0.241740i	$-0.0777182\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 4551.22i	− 0.428753i
$484$	0	0
$485$	−13260.7	−1.24152
$486$	0	0
$487$	13633.3i	1.26855i	0.773108	+	0.634275i	$0.218701\pi$
$487$	13633.3i	1.26855i	−0.773108	+	0.634275i	$0.781299\pi$
$488$	0	0
$489$	9954.12i	0.920533i
$490$	0	0
$491$	−664.188	−0.0610476	−0.0305238	−	0.999534i	$-0.509718\pi$
$491$	−664.188	−0.0610476	−0.0305238	+	0.999534i	$0.509718\pi$
$492$	0	0
$493$	−23552.0	−2.15158
$494$	0	0
$495$	7609.79	0.690979
$496$	0	0
$497$	24131.7	2.17797
$498$	0	0
$499$	14754.0i	1.32361i	0.749676	+	0.661805i	$0.230209\pi$
$499$	14754.0i	1.32361i	−0.749676	+	0.661805i	$0.769791\pi$
$500$	0	0
$501$	− 2548.55i	− 0.227267i
$502$	0	0
$503$	−5283.68	−0.468365	−0.234182	−	0.972193i	$-0.575241\pi$
$503$	−5283.68	−0.468365	−0.234182	+	0.972193i	$0.575241\pi$
$504$	0	0
$505$	14093.7i	1.24190i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7600.64i	0.661871i	0.943653	+	0.330936i	$0.107364\pi$
$509$	7600.64i	0.661871i	−0.943653	+	0.330936i	$0.892636\pi$
$510$	0	0
$511$	4935.84	0.427296
$512$	0	0
$513$	4521.97i	0.389181i
$514$	0	0
$515$	16151.0i	1.38194i
$516$	0	0
$517$	2886.09	0.245513
$518$	0	0
$519$	38902.8	3.29025
$520$	0	0
$521$	10834.5	0.911074	0.455537	−	0.890217i	$-0.349447\pi$
$521$	10834.5	0.911074	0.455537	+	0.890217i	$0.349447\pi$
$522$	0	0
$523$	9036.49	0.755522	0.377761	−	0.925903i	$-0.376694\pi$
$523$	9036.49	0.755522	0.377761	+	0.925903i	$0.376694\pi$
$524$	0	0
$525$	− 4173.51i	− 0.346946i
$526$	0	0
$527$	− 7542.35i	− 0.623434i
$528$	0	0
$529$	−11918.1	−0.979541
$530$	0	0
$531$	13186.3i	1.07766i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	20511.8i	1.65758i
$536$	0	0
$537$	−5832.41	−0.468691
$538$	0	0
$539$	8375.34i	0.669297i
$540$	0	0
$541$	− 8389.76i	− 0.666735i	−0.942797	−	0.333368i	$-0.891815\pi$
$541$	− 8389.76i	− 0.666735i	0.942797	−	0.333368i	$-0.108185\pi$
$542$	0	0
$543$	4070.02	0.321660
$544$	0	0
$545$	11177.4	0.878511
$546$	0	0
$547$	−23840.9	−1.86355	−0.931777	−	0.363032i	$-0.881742\pi$
$547$	−23840.9	−1.86355	−0.931777	+	0.363032i	$0.881742\pi$
$548$	0	0
$549$	10860.7	0.844307
$550$	0	0
$551$	5030.48i	0.388939i
$552$	0	0
$553$	− 28370.3i	− 2.18160i
$554$	0	0
$555$	28703.3	2.19529
$556$	0	0
$557$	− 18818.9i	− 1.43157i	−0.698322	−	0.715784i	$-0.746070\pi$
$557$	− 18818.9i	− 1.43157i	0.698322	−	0.715784i	$-0.253930\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	9721.60i	0.731634i
$562$	0	0
$563$	8552.08	0.640191	0.320095	−	0.947385i	$-0.396285\pi$
$563$	8552.08	0.640191	0.320095	+	0.947385i	$0.396285\pi$
$564$	0	0
$565$	− 11472.2i	− 0.854228i
$566$	0	0
$567$	21703.3i	1.60750i
$568$	0	0
$569$	−26223.7	−1.93208	−0.966041	−	0.258388i	$-0.916809\pi$
$569$	−26223.7	−1.93208	−0.966041	+	0.258388i	$0.916809\pi$
$570$	0	0
$571$	−3528.25	−0.258586	−0.129293	−	0.991606i	$-0.541271\pi$
$571$	−3528.25	−0.258586	−0.129293	+	0.991606i	$0.541271\pi$
$572$	0	0
$573$	−25230.6	−1.83948
$574$	0	0
$575$	228.267	0.0165555
$576$	0	0
$577$	− 26672.4i	− 1.92441i	−0.272322	−	0.962206i	$-0.587792\pi$
$577$	− 26672.4i	− 1.92441i	0.272322	−	0.962206i	$-0.412208\pi$
$578$	0	0
$579$	− 10338.4i	− 0.742054i
$580$	0	0
$581$	6341.41	0.452816
$582$	0	0
$583$	1139.36i	0.0809388i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4638.49i	0.326151i	0.986614	+	0.163076i	$0.0521415\pi$
$587$	4638.49i	0.326151i	−0.986614	+	0.163076i	$0.947859\pi$
$588$	0	0
$589$	−1610.97	−0.112697
$590$	0	0
$591$	− 3644.49i	− 0.253662i
$592$	0	0
$593$	5932.05i	0.410793i	0.978679	+	0.205396i	$0.0658484\pi$
$593$	5932.05i	0.410793i	−0.978679	+	0.205396i	$0.934152\pi$
$594$	0	0
$595$	34118.9	2.35082
$596$	0	0
$597$	7253.89	0.497289
$598$	0	0
$599$	−8927.94	−0.608991	−0.304496	−	0.952514i	$-0.598488\pi$
$599$	−8927.94	−0.608991	−0.304496	+	0.952514i	$0.598488\pi$
$600$	0	0
$601$	10434.5	0.708209	0.354104	−	0.935206i	$-0.384786\pi$
$601$	10434.5	0.708209	0.354104	+	0.935206i	$0.384786\pi$
$602$	0	0
$603$	− 12340.7i	− 0.833417i
$604$	0	0
$605$	13993.8i	0.940379i
$606$	0	0
$607$	8937.73	0.597646	0.298823	−	0.954309i	$-0.403406\pi$
$607$	8937.73	0.597646	0.298823	+	0.954309i	$0.403406\pi$
$608$	0	0
$609$	75691.5i	5.03641i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 23968.4i	− 1.57924i	−0.613597	−	0.789620i	$-0.710278\pi$
$613$	− 23968.4i	− 1.57924i	0.613597	−	0.789620i	$-0.289722\pi$
$614$	0	0
$615$	−17487.2	−1.14659
$616$	0	0
$617$	13104.0i	0.855017i	0.904011	+	0.427509i	$0.140609\pi$
$617$	13104.0i	0.855017i	−0.904011	+	0.427509i	$0.859391\pi$
$618$	0	0
$619$	− 25051.3i	− 1.62665i	−0.581810	−	0.813324i	$-0.697655\pi$
$619$	− 25051.3i	− 1.62665i	0.581810	−	0.813324i	$-0.302345\pi$
$620$	0	0
$621$	−3721.41	−0.240475
$622$	0	0
$623$	−44761.0	−2.87851
$624$	0	0
$625$	−17224.2	−1.10235
$626$	0	0
$627$	2076.44	0.132257
$628$	0	0
$629$	24342.2i	1.54306i
$630$	0	0
$631$	− 55.6934i	− 0.00351366i	−0.999998	−	0.00175683i	$-0.999441\pi$
$631$	− 55.6934i	− 0.00351366i	0.999998	−	0.00175683i	$-0.000559217\pi$
$632$	0	0
$633$	32268.3	2.02614
$634$	0	0
$635$	18941.0i	1.18370i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 39974.5i	− 2.47475i
$640$	0	0
$641$	17899.2	1.10293	0.551464	−	0.834199i	$-0.314069\pi$
$641$	17899.2	1.10293	0.551464	+	0.834199i	$0.314069\pi$
$642$	0	0
$643$	− 25880.4i	− 1.58729i	−0.608384	−	0.793643i	$-0.708182\pi$
$643$	− 25880.4i	− 1.58729i	0.608384	−	0.793643i	$-0.291818\pi$
$644$	0	0
$645$	− 28609.0i	− 1.74648i
$646$	0	0
$647$	137.706	0.00836749	0.00418375	−	0.999991i	$-0.498668\pi$
$647$	137.706	0.00836749	0.00418375	+	0.999991i	$0.498668\pi$
$648$	0	0
$649$	2988.81	0.180772
$650$	0	0
$651$	−24239.6	−1.45933
$652$	0	0
$653$	16095.2	0.964553	0.482277	−	0.876019i	$-0.339810\pi$
$653$	16095.2	0.964553	0.482277	+	0.876019i	$0.339810\pi$
$654$	0	0
$655$	19807.2i	1.18157i
$656$	0	0
$657$	− 8176.29i	− 0.485521i
$658$	0	0
$659$	−7055.14	−0.417040	−0.208520	−	0.978018i	$-0.566865\pi$
$659$	−7055.14	−0.417040	−0.208520	+	0.978018i	$0.566865\pi$
$660$	0	0
$661$	4042.63i	0.237882i	0.992901	+	0.118941i	$0.0379499\pi$
$661$	4042.63i	0.237882i	−0.992901	+	0.118941i	$0.962050\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 7287.46i	− 0.424956i
$666$	0	0
$667$	−4139.89	−0.240326
$668$	0	0
$669$	23398.2i	1.35221i
$670$	0	0
$671$	− 2461.70i	− 0.141629i
$672$	0	0
$673$	−22717.8	−1.30120	−0.650598	−	0.759422i	$-0.725482\pi$
$673$	−22717.8	−1.30120	−0.650598	+	0.759422i	$0.725482\pi$
$674$	0	0
$675$	−3412.57	−0.194592
$676$	0	0
$677$	19237.7	1.09212	0.546058	−	0.837747i	$-0.316128\pi$
$677$	19237.7	1.09212	0.546058	+	0.837747i	$0.316128\pi$
$678$	0	0
$679$	−36142.0	−2.04271
$680$	0	0
$681$	− 41771.0i	− 2.35047i
$682$	0	0
$683$	25082.9i	1.40523i	0.711571	+	0.702614i	$0.247984\pi$
$683$	25082.9i	1.40523i	−0.711571	+	0.702614i	$0.752016\pi$
$684$	0	0
$685$	6897.61	0.384736
$686$	0	0
$687$	1110.34i	0.0616624i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	11261.6i	0.619987i	0.950739	+	0.309994i	$0.100327\pi$
$691$	11261.6i	0.619987i	−0.950739	+	0.309994i	$0.899673\pi$
$692$	0	0
$693$	20740.5	1.13689
$694$	0	0
$695$	1849.51i	0.100944i
$696$	0	0
$697$	− 14830.2i	− 0.805933i
$698$	0	0
$699$	−10633.0	−0.575360
$700$	0	0
$701$	15678.7	0.844761	0.422381	−	0.906419i	$-0.361195\pi$
$701$	15678.7	0.844761	0.422381	+	0.906419i	$0.361195\pi$
$702$	0	0
$703$	5199.25	0.278938
$704$	0	0
$705$	−25275.5	−1.35026
$706$	0	0
$707$	38412.4i	2.04335i
$708$	0	0
$709$	− 17595.7i	− 0.932044i	−0.884773	−	0.466022i	$-0.845687\pi$
$709$	− 17595.7i	− 0.932044i	0.884773	−	0.466022i	$-0.154313\pi$
$710$	0	0
$711$	−46995.8	−2.47888
$712$	0	0
$713$	− 1325.77i	− 0.0696359i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	40719.7i	2.12093i
$718$	0	0
$719$	−16610.7	−0.861579	−0.430790	−	0.902452i	$-0.641765\pi$
$719$	−16610.7	−0.861579	−0.430790	+	0.902452i	$0.641765\pi$
$720$	0	0
$721$	44019.7i	2.27376i
$722$	0	0
$723$	− 56319.1i	− 2.89700i
$724$	0	0
$725$	−3796.32	−0.194471
$726$	0	0
$727$	8614.24	0.439456	0.219728	−	0.975561i	$-0.429483\pi$
$727$	8614.24	0.439456	0.219728	+	0.975561i	$0.429483\pi$
$728$	0	0
$729$	−21123.1	−1.07317
$730$	0	0
$731$	24262.2	1.22759
$732$	0	0
$733$	− 22282.4i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$733$	− 22282.4i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$734$	0	0
$735$	− 73348.7i	− 3.68096i
$736$	0	0
$737$	−2797.14	−0.139802
$738$	0	0
$739$	26990.0i	1.34350i	0.740780	+	0.671748i	$0.234456\pi$
$739$	26990.0i	1.34350i	−0.740780	+	0.671748i	$0.765544\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 9210.08i	− 0.454758i	−0.973806	−	0.227379i	$-0.926984\pi$
$743$	− 9210.08i	− 0.454758i	0.973806	−	0.227379i	$-0.0730156\pi$
$744$	0	0
$745$	−14018.2	−0.689381
$746$	0	0
$747$	− 10504.6i	− 0.514518i
$748$	0	0
$749$	55905.1i	2.72727i
$750$	0	0
$751$	−22231.6	−1.08022	−0.540109	−	0.841595i	$-0.681617\pi$
$751$	−22231.6	−1.08022	−0.540109	+	0.841595i	$0.681617\pi$
$752$	0	0
$753$	−8288.20	−0.401114
$754$	0	0
$755$	23882.5	1.15123
$756$	0	0
$757$	−21934.9	−1.05315	−0.526576	−	0.850128i	$-0.676525\pi$
$757$	−21934.9	−1.05315	−0.526576	+	0.850128i	$0.676525\pi$
$758$	0	0
$759$	1708.83i	0.0817215i
$760$	0	0
$761$	24239.1i	1.15462i	0.816524	+	0.577311i	$0.195898\pi$
$761$	24239.1i	1.15462i	−0.816524	+	0.577311i	$0.804102\pi$
$762$	0	0
$763$	30464.2	1.44545
$764$	0	0
$765$	− 56518.5i	− 2.67115i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 30974.8i	− 1.45251i	−0.687425	−	0.726256i	$-0.741259\pi$
$769$	− 30974.8i	− 1.45251i	0.687425	−	0.726256i	$-0.258741\pi$
$770$	0	0
$771$	4861.12	0.227067
$772$	0	0
$773$	− 32176.5i	− 1.49717i	−0.663041	−	0.748583i	$-0.730734\pi$
$773$	− 32176.5i	− 1.49717i	0.663041	−	0.748583i	$-0.269266\pi$
$774$	0	0
$775$	− 1215.74i	− 0.0563493i
$776$	0	0
$777$	78230.9	3.61199
$778$	0	0
$779$	−3167.59	−0.145688
$780$	0	0
$781$	−9060.63	−0.415128
$782$	0	0
$783$	61891.0	2.82478
$784$	0	0
$785$	16886.1i	0.767760i
$786$	0	0
$787$	26017.1i	1.17841i	0.807983	+	0.589206i	$0.200559\pi$
$787$	26017.1i	1.17841i	−0.807983	+	0.589206i	$0.799441\pi$
$788$	0	0
$789$	−47040.0	−2.12252
$790$	0	0
$791$	− 31267.5i	− 1.40549i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 9978.13i	− 0.445142i
$796$	0	0
$797$	18901.9	0.840076	0.420038	−	0.907507i	$-0.362017\pi$
$797$	18901.9	0.840076	0.420038	+	0.907507i	$0.362017\pi$
$798$	0	0
$799$	− 21435.3i	− 0.949092i
$800$	0	0
$801$	74147.2i	3.27074i
$802$	0	0
$803$	−1853.24	−0.0814439
$804$	0	0
$805$	5997.31	0.262580
$806$	0	0
$807$	−2958.92	−0.129069
$808$	0	0
$809$	727.574	0.0316195	0.0158097	−	0.999875i	$-0.494967\pi$
$809$	727.574	0.0316195	0.0158097	+	0.999875i	$0.494967\pi$
$810$	0	0
$811$	− 23940.1i	− 1.03656i	−0.855210	−	0.518281i	$-0.826572\pi$
$811$	− 23940.1i	− 1.03656i	0.855210	−	0.518281i	$-0.173428\pi$
$812$	0	0
$813$	− 12850.5i	− 0.554351i
$814$	0	0
$815$	−13116.9	−0.563761
$816$	0	0
$817$	− 5182.17i	− 0.221911i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	7561.37i	0.321430i	0.987001	+	0.160715i	$0.0513799\pi$
$821$	7561.37i	0.321430i	−0.987001	+	0.160715i	$0.948620\pi$
$822$	0	0
$823$	−20826.0	−0.882077	−0.441038	−	0.897488i	$-0.645390\pi$
$823$	−20826.0	−0.882077	−0.441038	+	0.897488i	$0.645390\pi$
$824$	0	0
$825$	1567.01i	0.0661289i
$826$	0	0
$827$	34592.6i	1.45454i	0.686353	+	0.727268i	$0.259210\pi$
$827$	34592.6i	1.45454i	−0.686353	+	0.727268i	$0.740790\pi$
$828$	0	0
$829$	20884.0	0.874948	0.437474	−	0.899231i	$-0.355873\pi$
$829$	20884.0	0.874948	0.437474	+	0.899231i	$0.355873\pi$
$830$	0	0
$831$	19188.5	0.801011
$832$	0	0
$833$	62204.3	2.58734
$834$	0	0
$835$	3358.32	0.139185
$836$	0	0
$837$	19820.1i	0.818498i
$838$	0	0
$839$	43983.4i	1.80986i	0.425557	+	0.904932i	$0.360078\pi$
$839$	43983.4i	1.80986i	−0.425557	+	0.904932i	$0.639922\pi$
$840$	0	0
$841$	44461.8	1.82303
$842$	0	0
$843$	− 52491.2i	− 2.14459i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	38140.2i	1.54724i
$848$	0	0
$849$	15302.7	0.618595
$850$	0	0
$851$	4278.79i	0.172356i
$852$	0	0
$853$	− 16552.4i	− 0.664412i	−0.943207	−	0.332206i	$-0.892207\pi$
$853$	− 16552.4i	− 0.664412i	0.943207	−	0.332206i	$-0.107793\pi$
$854$	0	0
$855$	−12071.8	−0.482861
$856$	0	0
$857$	−12869.9	−0.512983	−0.256492	−	0.966546i	$-0.582567\pi$
$857$	−12869.9	−0.512983	−0.256492	+	0.966546i	$0.582567\pi$
$858$	0	0
$859$	21172.2	0.840961	0.420481	−	0.907302i	$-0.361861\pi$
$859$	21172.2	0.840961	0.420481	+	0.907302i	$0.361861\pi$
$860$	0	0
$861$	−47661.4	−1.88652
$862$	0	0
$863$	− 47889.1i	− 1.88895i	−0.328584	−	0.944475i	$-0.606571\pi$
$863$	− 47889.1i	− 1.88895i	0.328584	−	0.944475i	$-0.393429\pi$
$864$	0	0
$865$	51263.6i	2.01505i
$866$	0	0
$867$	28172.5	1.10356
$868$	0	0
$869$	10652.1i	0.415820i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	59869.7i	2.32106i
$874$	0	0
$875$	−42015.5	−1.62330
$876$	0	0
$877$	43584.8i	1.67817i	0.544000	+	0.839085i	$0.316909\pi$
$877$	43584.8i	1.67817i	−0.544000	+	0.839085i	$0.683091\pi$
$878$	0	0
$879$	− 15981.9i	− 0.613262i
$880$	0	0
$881$	−42558.4	−1.62750	−0.813751	−	0.581214i	$-0.802578\pi$
$881$	−42558.4	−1.62750	−0.813751	+	0.581214i	$0.802578\pi$
$882$	0	0
$883$	−22522.3	−0.858364	−0.429182	−	0.903218i	$-0.641198\pi$
$883$	−22522.3	−0.858364	−0.429182	+	0.903218i	$0.641198\pi$
$884$	0	0
$885$	−26175.1	−0.994198
$886$	0	0
$887$	−6178.80	−0.233894	−0.116947	−	0.993138i	$-0.537311\pi$
$887$	−6178.80	−0.233894	−0.116947	+	0.993138i	$0.537311\pi$
$888$	0	0
$889$	51623.9i	1.94759i
$890$	0	0
$891$	− 8148.85i	− 0.306394i
$892$	0	0
$893$	−4578.35	−0.171566
$894$	0	0
$895$	− 7685.59i	− 0.287040i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	22048.9i	0.817989i
$900$	0	0
$901$	8462.09	0.312889
$902$	0	0
$903$	− 77974.0i	− 2.87354i
$904$	0	0
$905$	5363.22i	0.196994i
$906$	0	0
$907$	−26155.5	−0.957528	−0.478764	−	0.877944i	$-0.658915\pi$
$907$	−26155.5	−0.957528	−0.478764	+	0.877944i	$0.658915\pi$
$908$	0	0
$909$	63630.8	2.32178
$910$	0	0
$911$	29386.6	1.06874	0.534369	−	0.845251i	$-0.320549\pi$
$911$	29386.6	1.06874	0.534369	+	0.845251i	$0.320549\pi$
$912$	0	0
$913$	−2380.99	−0.0863080
$914$	0	0
$915$	21558.8i	0.778920i
$916$	0	0
$917$	53984.6i	1.94409i
$918$	0	0
$919$	22697.8	0.814726	0.407363	−	0.913266i	$-0.366448\pi$
$919$	22697.8	0.814726	0.407363	+	0.913266i	$0.366448\pi$
$920$	0	0
$921$	− 36097.9i	− 1.29149i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	3923.69i	0.139470i
$926$	0	0
$927$	72919.2	2.58358
$928$	0	0
$929$	40792.6i	1.44065i	0.693638	+	0.720324i	$0.256007\pi$
$929$	40792.6i	1.44065i	−0.693638	+	0.720324i	$0.743993\pi$
$930$	0	0
$931$	− 13286.2i	− 0.467710i
$932$	0	0
$933$	−722.619	−0.0253563
$934$	0	0
$935$	−12810.5	−0.448074
$936$	0	0
$937$	22015.2	0.767560	0.383780	−	0.923424i	$-0.374622\pi$
$937$	22015.2	0.767560	0.383780	+	0.923424i	$0.374622\pi$
$938$	0	0
$939$	−41478.4	−1.44153
$940$	0	0
$941$	− 7774.05i	− 0.269316i	−0.990892	−	0.134658i	$-0.957006\pi$
$941$	− 7774.05i	− 0.269316i	0.990892	−	0.134658i	$-0.0429936\pi$
$942$	0	0
$943$	− 2606.81i	− 0.0900204i
$944$	0	0
$945$	−89659.2	−3.08636
$946$	0	0
$947$	15872.9i	0.544666i	0.962203	+	0.272333i	$0.0877953\pi$
$947$	15872.9i	0.544666i	−0.962203	+	0.272333i	$0.912205\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 96102.6i	− 3.27691i
$952$	0	0
$953$	−43815.4	−1.48932	−0.744659	−	0.667445i	$-0.767388\pi$
$953$	−43815.4	−1.48932	−0.744659	+	0.667445i	$0.767388\pi$
$954$	0	0
$955$	− 33247.3i	− 1.12655i
$956$	0	0
$957$	− 28419.6i	− 0.959955i
$958$	0	0
$959$	18799.5	0.633020
$960$	0	0
$961$	22730.0	0.762983
$962$	0	0
$963$	92607.6	3.09890
$964$	0	0
$965$	13623.3	0.454455
$966$	0	0
$967$	− 6323.57i	− 0.210292i	−0.994457	−	0.105146i	$-0.966469\pi$
$967$	− 6323.57i	− 0.210292i	0.994457	−	0.105146i	$-0.0335310\pi$
$968$	0	0
$969$	− 15421.9i	− 0.511271i
$970$	0	0
$971$	−49510.7	−1.63633	−0.818164	−	0.574985i	$-0.805008\pi$
$971$	−49510.7	−1.63633	−0.818164	+	0.574985i	$0.805008\pi$
$972$	0	0
$973$	5040.85i	0.166087i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5655.15i	0.185184i	0.995704	+	0.0925918i	$0.0295152\pi$
$977$	5655.15i	0.185184i	−0.995704	+	0.0925918i	$0.970485\pi$
$978$	0	0
$979$	16806.2	0.548652
$980$	0	0
$981$	− 50464.3i	− 1.64241i
$982$	0	0
$983$	− 4051.62i	− 0.131461i	−0.997837	−	0.0657307i	$-0.979062\pi$
$983$	− 4051.62i	− 0.131461i	0.997837	−	0.0657307i	$-0.0209378\pi$
$984$	0	0
$985$	4802.48	0.155350
$986$	0	0
$987$	−68888.6	−2.22163
$988$	0	0
$989$	4264.73	0.137119
$990$	0	0
$991$	−20208.3	−0.647768	−0.323884	−	0.946097i	$-0.604989\pi$
$991$	−20208.3	−0.647768	−0.323884	+	0.946097i	$0.604989\pi$
$992$	0	0
$993$	− 46535.7i	− 1.48718i
$994$	0	0
$995$	9558.72i	0.304554i
$996$	0	0
$997$	−46278.3	−1.47006	−0.735029	−	0.678035i	$-0.762832\pi$
$997$	−46278.3	−1.47006	−0.735029	+	0.678035i	$0.762832\pi$
$998$	0	0
$999$	− 63967.4i	− 2.02587i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	676.4.d.d.337.8		8
13.2	odd	12		676.4.e.h.529.2		16
13.3	even	3		676.4.h.e.485.1		8
13.4	even	6		676.4.h.e.361.1		8
13.5	odd	4		676.4.a.g.1.8		8
13.6	odd	12		676.4.e.h.653.2		16
13.7	odd	12		676.4.e.h.653.1		16
13.8	odd	4		676.4.a.g.1.7		8
13.9	even	3		52.4.h.a.49.1	yes	8
13.10	even	6		52.4.h.a.17.1	✓	8
13.11	odd	12		676.4.e.h.529.1		16
13.12	even	2	inner	676.4.d.d.337.7		8
39.23	odd	6		468.4.t.g.433.3		8
39.35	odd	6		468.4.t.g.361.2		8
52.23	odd	6		208.4.w.c.17.4		8
52.35	odd	6		208.4.w.c.49.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.4.h.a.17.1	✓	8	13.10	even	6
52.4.h.a.49.1	yes	8	13.9	even	3
208.4.w.c.17.4		8	52.23	odd	6
208.4.w.c.49.4		8	52.35	odd	6
468.4.t.g.361.2		8	39.35	odd	6
468.4.t.g.433.3		8	39.23	odd	6
676.4.a.g.1.7		8	13.8	odd	4
676.4.a.g.1.8		8	13.5	odd	4
676.4.d.d.337.7		8	13.12	even	2	inner
676.4.d.d.337.8		8	1.1	even	1	trivial
676.4.e.h.529.1		16	13.11	odd	12
676.4.e.h.529.2		16	13.2	odd	12
676.4.e.h.653.1		16	13.7	odd	12
676.4.e.h.653.2		16	13.6	odd	12
676.4.h.e.361.1		8	13.4	even	6
676.4.h.e.485.1		8	13.3	even	3

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:	\( N \)	\(=\)	\( 676 = 2^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	676.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.96207 q^{3} +11.8097i q^{5} +32.1873i q^{7} +53.3187 q^{9} +O(q^{10})\)	\(q+8.96207 q^{3} +11.8097i q^{5} +32.1873i q^{7} +53.3187 q^{9} -12.0853i q^{11} +105.839i q^{15} -89.7581 q^{17} +19.1714i q^{19} +288.465i q^{21} -15.7774 q^{23} -14.4680 q^{25} +235.870 q^{27} +262.394 q^{29} +84.0297i q^{31} -108.309i q^{33} -380.121 q^{35} -271.198i q^{37} +165.224i q^{41} -270.307 q^{43} +629.676i q^{45} +238.811i q^{47} -693.021 q^{49} -804.419 q^{51} -94.2765 q^{53} +142.723 q^{55} +171.816i q^{57} +247.310i q^{59} +203.694 q^{61} +1716.18i q^{63} -231.451i q^{67} -141.398 q^{69} -749.727i q^{71} -153.347i q^{73} -129.663 q^{75} +388.991 q^{77} -881.413 q^{79} +674.281 q^{81} -197.016i q^{83} -1060.01i q^{85} +2351.60 q^{87} +1390.64i q^{89} +753.080i q^{93} -226.408 q^{95} +1122.87i q^{97} -644.370i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 140 q^{9}+O(q^{10}) \) 8 * q + 140 * q^9	\( 8 q + 140 q^{9} - 176 q^{17} + 40 q^{23} - 84 q^{25} - 432 q^{27} + 968 q^{29} - 80 q^{35} - 1008 q^{43} - 1844 q^{49} - 1808 q^{51} - 1164 q^{53} + 2256 q^{55} + 2448 q^{61} - 3476 q^{69} - 2896 q^{75} - 3972 q^{77} - 3968 q^{79} + 8264 q^{81} + 3320 q^{87} - 4800 q^{95}+O(q^{100}) \) 8 * q + 140 * q^9 - 176 * q^17 + 40 * q^23 - 84 * q^25 - 432 * q^27 + 968 * q^29 - 80 * q^35 - 1008 * q^43 - 1844 * q^49 - 1808 * q^51 - 1164 * q^53 + 2256 * q^55 + 2448 * q^61 - 3476 * q^69 - 2896 * q^75 - 3972 * q^77 - 3968 * q^79 + 8264 * q^81 + 3320 * q^87 - 4800 * q^95

Introduction

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