LMFDB - Embedded newform 312.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [312,4,Mod(1,312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(312, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("312.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 312 = 2^{3} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	312.a (trivial)

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:	yes
Analytic conductor:	$18.4085959218$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{55}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 55 $ x^2 - 55
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-7.41620$ of defining polynomial
Character	$\chi$	$=$	312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -16.8324 q^{5} -4.83240 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -16.8324 q^{5} -4.83240 q^{7} +9.00000 q^{9} -39.6648 q^{11} -13.0000 q^{13} +50.4972 q^{15} -4.33521 q^{17} +28.8324 q^{19} +14.4972 q^{21} +0.670412 q^{23} +158.330 q^{25} -27.0000 q^{27} -6.00000 q^{29} +274.497 q^{31} +118.994 q^{33} +81.3408 q^{35} +84.3352 q^{37} +39.0000 q^{39} +209.156 q^{41} +364.324 q^{43} -151.492 q^{45} +239.341 q^{47} -319.648 q^{49} +13.0056 q^{51} -415.318 q^{53} +667.654 q^{55} -86.4972 q^{57} +78.0000 q^{59} -813.955 q^{61} -43.4916 q^{63} +218.821 q^{65} +858.452 q^{67} -2.01124 q^{69} -213.006 q^{71} +568.313 q^{73} -474.989 q^{75} +191.676 q^{77} +583.352 q^{79} +81.0000 q^{81} +162.648 q^{83} +72.9719 q^{85} +18.0000 q^{87} +1417.41 q^{89} +62.8212 q^{91} -823.492 q^{93} -485.318 q^{95} -798.949 q^{97} -356.983 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 4 q^{5} + 20 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 - 4 * q^5 + 20 * q^7 + 18 * q^9	$ 2 q - 6 q^{3} - 4 q^{5} + 20 q^{7} + 18 q^{9} - 20 q^{11} - 26 q^{13} + 12 q^{15} - 68 q^{17} + 28 q^{19} - 60 q^{21} + 120 q^{23} + 198 q^{25} - 54 q^{27} - 12 q^{29} + 460 q^{31} + 60 q^{33} + 400 q^{35} + 228 q^{37} + 78 q^{39} + 92 q^{41} + 432 q^{43} - 36 q^{45} + 716 q^{47} - 46 q^{49} + 204 q^{51} - 356 q^{53} + 920 q^{55} - 84 q^{57} + 156 q^{59} - 204 q^{61} + 180 q^{63} + 52 q^{65} + 204 q^{67} - 360 q^{69} - 604 q^{71} + 484 q^{73} - 594 q^{75} + 680 q^{77} + 1760 q^{79} + 162 q^{81} - 268 q^{83} - 744 q^{85} + 36 q^{87} + 76 q^{89} - 260 q^{91} - 1380 q^{93} - 496 q^{95} + 4 q^{97} - 180 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 4 * q^5 + 20 * q^7 + 18 * q^9 - 20 * q^11 - 26 * q^13 + 12 * q^15 - 68 * q^17 + 28 * q^19 - 60 * q^21 + 120 * q^23 + 198 * q^25 - 54 * q^27 - 12 * q^29 + 460 * q^31 + 60 * q^33 + 400 * q^35 + 228 * q^37 + 78 * q^39 + 92 * q^41 + 432 * q^43 - 36 * q^45 + 716 * q^47 - 46 * q^49 + 204 * q^51 - 356 * q^53 + 920 * q^55 - 84 * q^57 + 156 * q^59 - 204 * q^61 + 180 * q^63 + 52 * q^65 + 204 * q^67 - 360 * q^69 - 604 * q^71 + 484 * q^73 - 594 * q^75 + 680 * q^77 + 1760 * q^79 + 162 * q^81 - 268 * q^83 - 744 * q^85 + 36 * q^87 + 76 * q^89 - 260 * q^91 - 1380 * q^93 - 496 * q^95 + 4 * q^97 - 180 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−16.8324	−1.50554	−0.752768	−	0.658286i	$-0.771282\pi$
$5$	−16.8324	−1.50554	−0.752768	+	0.658286i	$0.771282\pi$
$6$	0	0
$7$	−4.83240	−0.260925	−0.130462	−	0.991453i	$-0.541646\pi$
$7$	−4.83240	−0.260925	−0.130462	+	0.991453i	$0.541646\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−39.6648	−1.08722	−0.543608	−	0.839339i	$-0.682942\pi$
$11$	−39.6648	−1.08722	−0.543608	+	0.839339i	$0.682942\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	50.4972	0.869221
$16$	0	0
$17$	−4.33521	−0.0618495	−0.0309248	−	0.999522i	$-0.509845\pi$
$17$	−4.33521	−0.0618495	−0.0309248	+	0.999522i	$0.509845\pi$
$18$	0	0
$19$	28.8324	0.348137	0.174069	−	0.984734i	$-0.444309\pi$
$19$	28.8324	0.348137	0.174069	+	0.984734i	$0.444309\pi$
$20$	0	0
$21$	14.4972	0.150645
$22$	0	0
$23$	0.670412	0.00607785	0.00303893	−	0.999995i	$-0.499033\pi$
$23$	0.670412	0.00607785	0.00303893	+	0.999995i	$0.499033\pi$
$24$	0	0
$25$	158.330	1.26664
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−6.00000	−0.0384197	−0.0192099	−	0.999815i	$-0.506115\pi$
$29$	−6.00000	−0.0384197	−0.0192099	+	0.999815i	$0.506115\pi$
$30$	0	0
$31$	274.497	1.59036	0.795180	−	0.606374i	$-0.207376\pi$
$31$	274.497	1.59036	0.795180	+	0.606374i	$0.207376\pi$
$32$	0	0
$33$	118.994	0.627705
$34$	0	0
$35$	81.3408	0.392832
$36$	0	0
$37$	84.3352	0.374720	0.187360	−	0.982291i	$-0.440007\pi$
$37$	84.3352	0.374720	0.187360	+	0.982291i	$0.440007\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	209.156	0.796701	0.398350	−	0.917233i	$-0.369583\pi$
$41$	209.156	0.796701	0.398350	+	0.917233i	$0.369583\pi$
$42$	0	0
$43$	364.324	1.29207	0.646034	−	0.763309i	$-0.276427\pi$
$43$	364.324	1.29207	0.646034	+	0.763309i	$0.276427\pi$
$44$	0	0
$45$	−151.492	−0.501845
$46$	0	0
$47$	239.341	0.742797	0.371398	−	0.928474i	$-0.378878\pi$
$47$	239.341	0.742797	0.371398	+	0.928474i	$0.378878\pi$
$48$	0	0
$49$	−319.648	−0.931918
$50$	0	0
$51$	13.0056	0.0357088
$52$	0	0
$53$	−415.318	−1.07638	−0.538192	−	0.842822i	$-0.680893\pi$
$53$	−415.318	−1.07638	−0.538192	+	0.842822i	$0.680893\pi$
$54$	0	0
$55$	667.654	1.63684
$56$	0	0
$57$	−86.4972	−0.200997
$58$	0	0
$59$	78.0000	0.172114	0.0860571	−	0.996290i	$-0.472573\pi$
$59$	78.0000	0.172114	0.0860571	+	0.996290i	$0.472573\pi$
$60$	0	0
$61$	−813.955	−1.70846	−0.854232	−	0.519893i	$-0.825972\pi$
$61$	−813.955	−1.70846	−0.854232	+	0.519893i	$0.825972\pi$
$62$	0	0
$63$	−43.4916	−0.0869750
$64$	0	0
$65$	218.821	0.417560
$66$	0	0
$67$	858.452	1.56532	0.782661	−	0.622448i	$-0.213862\pi$
$67$	858.452	1.56532	0.782661	+	0.622448i	$0.213862\pi$
$68$	0	0
$69$	−2.01124	−0.00350905
$70$	0	0
$71$	−213.006	−0.356044	−0.178022	−	0.984027i	$-0.556970\pi$
$71$	−213.006	−0.356044	−0.178022	+	0.984027i	$0.556970\pi$
$72$	0	0
$73$	568.313	0.911178	0.455589	−	0.890190i	$-0.349429\pi$
$73$	568.313	0.911178	0.455589	+	0.890190i	$0.349429\pi$
$74$	0	0
$75$	−474.989	−0.731293
$76$	0	0
$77$	191.676	0.283682
$78$	0	0
$79$	583.352	0.830788	0.415394	−	0.909642i	$-0.363644\pi$
$79$	583.352	0.830788	0.415394	+	0.909642i	$0.363644\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	162.648	0.215096	0.107548	−	0.994200i	$-0.465700\pi$
$83$	162.648	0.215096	0.107548	+	0.994200i	$0.465700\pi$
$84$	0	0
$85$	72.9719	0.0931166
$86$	0	0
$87$	18.0000	0.0221816
$88$	0	0
$89$	1417.41	1.68815	0.844076	−	0.536224i	$-0.180150\pi$
$89$	1417.41	1.68815	0.844076	+	0.536224i	$0.180150\pi$
$90$	0	0
$91$	62.8212	0.0723675
$92$	0	0
$93$	−823.492	−0.918195
$94$	0	0
$95$	−485.318	−0.524133
$96$	0	0
$97$	−798.949	−0.836299	−0.418150	−	0.908378i	$-0.637321\pi$
$97$	−798.949	−0.836299	−0.418150	+	0.908378i	$0.637321\pi$
$98$	0	0
$99$	−356.983	−0.362406
$100$	0	0
$101$	−1629.63	−1.60549	−0.802744	−	0.596323i	$-0.796628\pi$
$101$	−1629.63	−1.60549	−0.802744	+	0.596323i	$0.796628\pi$
$102$	0	0
$103$	−384.927	−0.368233	−0.184116	−	0.982904i	$-0.558942\pi$
$103$	−384.927	−0.368233	−0.184116	+	0.982904i	$0.558942\pi$
$104$	0	0
$105$	−244.022	−0.226801
$106$	0	0
$107$	−336.045	−0.303614	−0.151807	−	0.988410i	$-0.548509\pi$
$107$	−336.045	−0.303614	−0.151807	+	0.988410i	$0.548509\pi$
$108$	0	0
$109$	777.307	0.683051	0.341525	−	0.939873i	$-0.389056\pi$
$109$	777.307	0.683051	0.341525	+	0.939873i	$0.389056\pi$
$110$	0	0
$111$	−253.006	−0.216344
$112$	0	0
$113$	−338.648	−0.281923	−0.140962	−	0.990015i	$-0.545019\pi$
$113$	−338.648	−0.281923	−0.140962	+	0.990015i	$0.545019\pi$
$114$	0	0
$115$	−11.2846	−0.00915042
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	20.9494	0.0161381
$120$	0	0
$121$	242.296	0.182040
$122$	0	0
$123$	−627.469	−0.459975
$124$	0	0
$125$	−561.017	−0.401431
$126$	0	0
$127$	−1389.60	−0.970920	−0.485460	−	0.874259i	$-0.661348\pi$
$127$	−1389.60	−0.970920	−0.485460	+	0.874259i	$0.661348\pi$
$128$	0	0
$129$	−1092.97	−0.745975
$130$	0	0
$131$	−701.966	−0.468176	−0.234088	−	0.972215i	$-0.575210\pi$
$131$	−701.966	−0.468176	−0.234088	+	0.972215i	$0.575210\pi$
$132$	0	0
$133$	−139.330	−0.0908377
$134$	0	0
$135$	454.475	0.289740
$136$	0	0
$137$	−1044.21	−0.651187	−0.325593	−	0.945510i	$-0.605564\pi$
$137$	−1044.21	−0.651187	−0.325593	+	0.945510i	$0.605564\pi$
$138$	0	0
$139$	1935.33	1.18095	0.590477	−	0.807055i	$-0.298940\pi$
$139$	1935.33	1.18095	0.590477	+	0.807055i	$0.298940\pi$
$140$	0	0
$141$	−718.022	−0.428854
$142$	0	0
$143$	515.642	0.301540
$144$	0	0
$145$	100.994	0.0578423
$146$	0	0
$147$	958.944	0.538043
$148$	0	0
$149$	−2252.37	−1.23840	−0.619200	−	0.785233i	$-0.712543\pi$
$149$	−2252.37	−1.23840	−0.619200	+	0.785233i	$0.712543\pi$
$150$	0	0
$151$	626.128	0.337441	0.168721	−	0.985664i	$-0.446036\pi$
$151$	626.128	0.337441	0.168721	+	0.985664i	$0.446036\pi$
$152$	0	0
$153$	−39.0169	−0.0206165
$154$	0	0
$155$	−4620.45	−2.39434
$156$	0	0
$157$	1102.56	0.560470	0.280235	−	0.959931i	$-0.409588\pi$
$157$	1102.56	0.560470	0.280235	+	0.959931i	$0.409588\pi$
$158$	0	0
$159$	1245.96	0.621451
$160$	0	0
$161$	−3.23970	−0.00158586
$162$	0	0
$163$	695.145	0.334037	0.167018	−	0.985954i	$-0.446586\pi$
$163$	695.145	0.334037	0.167018	+	0.985954i	$0.446586\pi$
$164$	0	0
$165$	−2002.96	−0.945032
$166$	0	0
$167$	1984.50	0.919553	0.459777	−	0.888035i	$-0.347930\pi$
$167$	1984.50	0.919553	0.459777	+	0.888035i	$0.347930\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	259.492	0.116046
$172$	0	0
$173$	−1100.85	−0.483792	−0.241896	−	0.970302i	$-0.577769\pi$
$173$	−1100.85	−0.483792	−0.241896	+	0.970302i	$0.577769\pi$
$174$	0	0
$175$	−765.111	−0.330497
$176$	0	0
$177$	−234.000	−0.0993702
$178$	0	0
$179$	−3792.63	−1.58365	−0.791827	−	0.610745i	$-0.790870\pi$
$179$	−3792.63	−1.58365	−0.791827	+	0.610745i	$0.790870\pi$
$180$	0	0
$181$	1926.47	0.791123	0.395561	−	0.918440i	$-0.370550\pi$
$181$	1926.47	0.791123	0.395561	+	0.918440i	$0.370550\pi$
$182$	0	0
$183$	2441.87	0.986382
$184$	0	0
$185$	−1419.56	−0.564153
$186$	0	0
$187$	171.955	0.0672438
$188$	0	0
$189$	130.475	0.0502150
$190$	0	0
$191$	2048.51	0.776048	0.388024	−	0.921649i	$-0.373158\pi$
$191$	2048.51	0.776048	0.388024	+	0.921649i	$0.373158\pi$
$192$	0	0
$193$	4134.67	1.54207	0.771036	−	0.636791i	$-0.219739\pi$
$193$	4134.67	1.54207	0.771036	+	0.636791i	$0.219739\pi$
$194$	0	0
$195$	−656.463	−0.241079
$196$	0	0
$197$	1856.98	0.671594	0.335797	−	0.941934i	$-0.390994\pi$
$197$	1856.98	0.671594	0.335797	+	0.941934i	$0.390994\pi$
$198$	0	0
$199$	5515.56	1.96476	0.982382	−	0.186885i	$-0.0598392\pi$
$199$	5515.56	1.96476	0.982382	+	0.186885i	$0.0598392\pi$
$200$	0	0
$201$	−2575.36	−0.903740
$202$	0	0
$203$	28.9944	0.0100247
$204$	0	0
$205$	−3520.60	−1.19946
$206$	0	0
$207$	6.03371	0.00202595
$208$	0	0
$209$	−1143.63	−0.378501
$210$	0	0
$211$	4035.73	1.31674	0.658368	−	0.752697i	$-0.271247\pi$
$211$	4035.73	1.31674	0.658368	+	0.752697i	$0.271247\pi$
$212$	0	0
$213$	639.017	0.205562
$214$	0	0
$215$	−6132.45	−1.94525
$216$	0	0
$217$	−1326.48	−0.414964
$218$	0	0
$219$	−1704.94	−0.526069
$220$	0	0
$221$	56.3577	0.0171540
$222$	0	0
$223$	−4452.72	−1.33711	−0.668556	−	0.743661i	$-0.733088\pi$
$223$	−4452.72	−1.33711	−0.668556	+	0.743661i	$0.733088\pi$
$224$	0	0
$225$	1424.97	0.422212
$226$	0	0
$227$	5482.68	1.60308	0.801538	−	0.597944i	$-0.204015\pi$
$227$	5482.68	1.60308	0.801538	+	0.597944i	$0.204015\pi$
$228$	0	0
$229$	−1406.74	−0.405938	−0.202969	−	0.979185i	$-0.565059\pi$
$229$	−1406.74	−0.405938	−0.202969	+	0.979185i	$0.565059\pi$
$230$	0	0
$231$	−575.028	−0.163784
$232$	0	0
$233$	6420.16	1.80514	0.902572	−	0.430540i	$-0.141677\pi$
$233$	6420.16	1.80514	0.902572	+	0.430540i	$0.141677\pi$
$234$	0	0
$235$	−4028.68	−1.11831
$236$	0	0
$237$	−1750.06	−0.479656
$238$	0	0
$239$	−4899.44	−1.32602	−0.663009	−	0.748611i	$-0.730721\pi$
$239$	−4899.44	−1.32602	−0.663009	+	0.748611i	$0.730721\pi$
$240$	0	0
$241$	259.129	0.0692613	0.0346307	−	0.999400i	$-0.488975\pi$
$241$	259.129	0.0692613	0.0346307	+	0.999400i	$0.488975\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	5380.44	1.40304
$246$	0	0
$247$	−374.821	−0.0965559
$248$	0	0
$249$	−487.944	−0.124185
$250$	0	0
$251$	884.760	0.222492	0.111246	−	0.993793i	$-0.464516\pi$
$251$	884.760	0.222492	0.111246	+	0.993793i	$0.464516\pi$
$252$	0	0
$253$	−26.5918	−0.00660794
$254$	0	0
$255$	−218.916	−0.0537609
$256$	0	0
$257$	2913.01	0.707036	0.353518	−	0.935428i	$-0.384985\pi$
$257$	2913.01	0.707036	0.353518	+	0.935428i	$0.384985\pi$
$258$	0	0
$259$	−407.541	−0.0977737
$260$	0	0
$261$	−54.0000	−0.0128066
$262$	0	0
$263$	−127.330	−0.0298535	−0.0149268	−	0.999889i	$-0.504752\pi$
$263$	−127.330	−0.0298535	−0.0149268	+	0.999889i	$0.504752\pi$
$264$	0	0
$265$	6990.80	1.62053
$266$	0	0
$267$	−4252.24	−0.974655
$268$	0	0
$269$	8177.51	1.85350	0.926750	−	0.375679i	$-0.122591\pi$
$269$	8177.51	1.85350	0.926750	+	0.375679i	$0.122591\pi$
$270$	0	0
$271$	−4113.84	−0.922132	−0.461066	−	0.887366i	$-0.652533\pi$
$271$	−4113.84	−0.922132	−0.461066	+	0.887366i	$0.652533\pi$
$272$	0	0
$273$	−188.463	−0.0417814
$274$	0	0
$275$	−6280.11	−1.37711
$276$	0	0
$277$	587.543	0.127444	0.0637221	−	0.997968i	$-0.479703\pi$
$277$	587.543	0.127444	0.0637221	+	0.997968i	$0.479703\pi$
$278$	0	0
$279$	2470.47	0.530120
$280$	0	0
$281$	3925.13	0.833288	0.416644	−	0.909070i	$-0.363206\pi$
$281$	3925.13	0.833288	0.416644	+	0.909070i	$0.363206\pi$
$282$	0	0
$283$	6461.26	1.35718	0.678590	−	0.734517i	$-0.262591\pi$
$283$	6461.26	1.35718	0.678590	+	0.734517i	$0.262591\pi$
$284$	0	0
$285$	1455.96	0.302608
$286$	0	0
$287$	−1010.73	−0.207879
$288$	0	0
$289$	−4894.21	−0.996175
$290$	0	0
$291$	2396.85	0.482838
$292$	0	0
$293$	9603.82	1.91489	0.957443	−	0.288623i	$-0.0931975\pi$
$293$	9603.82	1.91489	0.957443	+	0.288623i	$0.0931975\pi$
$294$	0	0
$295$	−1312.93	−0.259124
$296$	0	0
$297$	1070.95	0.209235
$298$	0	0
$299$	−8.71536	−0.00168569
$300$	0	0
$301$	−1760.56	−0.337132
$302$	0	0
$303$	4888.89	0.926929
$304$	0	0
$305$	13700.8	2.57215
$306$	0	0
$307$	5313.21	0.987756	0.493878	−	0.869531i	$-0.335579\pi$
$307$	5313.21	0.987756	0.493878	+	0.869531i	$0.335579\pi$
$308$	0	0
$309$	1154.78	0.212599
$310$	0	0
$311$	6662.88	1.21485	0.607423	−	0.794378i	$-0.292203\pi$
$311$	6662.88	1.21485	0.607423	+	0.794378i	$0.292203\pi$
$312$	0	0
$313$	1556.70	0.281119	0.140559	−	0.990072i	$-0.455110\pi$
$313$	1556.70	0.281119	0.140559	+	0.990072i	$0.455110\pi$
$314$	0	0
$315$	732.067	0.130944
$316$	0	0
$317$	−662.664	−0.117410	−0.0587049	−	0.998275i	$-0.518697\pi$
$317$	−662.664	−0.117410	−0.0587049	+	0.998275i	$0.518697\pi$
$318$	0	0
$319$	237.989	0.0417706
$320$	0	0
$321$	1008.13	0.175292
$322$	0	0
$323$	−124.994	−0.0215321
$324$	0	0
$325$	−2058.28	−0.351302
$326$	0	0
$327$	−2331.92	−0.394359
$328$	0	0
$329$	−1156.59	−0.193814
$330$	0	0
$331$	1378.83	0.228965	0.114482	−	0.993425i	$-0.463479\pi$
$331$	1378.83	0.228965	0.114482	+	0.993425i	$0.463479\pi$
$332$	0	0
$333$	759.017	0.124907
$334$	0	0
$335$	−14449.8	−2.35665
$336$	0	0
$337$	8180.86	1.32237	0.661187	−	0.750222i	$-0.270053\pi$
$337$	8180.86	1.32237	0.661187	+	0.750222i	$0.270053\pi$
$338$	0	0
$339$	1015.94	0.162768
$340$	0	0
$341$	−10887.9	−1.72907
$342$	0	0
$343$	3202.18	0.504086
$344$	0	0
$345$	33.8539	0.00528300
$346$	0	0
$347$	2712.49	0.419637	0.209819	−	0.977740i	$-0.432713\pi$
$347$	2712.49	0.419637	0.209819	+	0.977740i	$0.432713\pi$
$348$	0	0
$349$	−6789.20	−1.04131	−0.520656	−	0.853767i	$-0.674313\pi$
$349$	−6789.20	−1.04131	−0.520656	+	0.853767i	$0.674313\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	228.018	0.0343801	0.0171900	−	0.999852i	$-0.494528\pi$
$353$	228.018	0.0343801	0.0171900	+	0.999852i	$0.494528\pi$
$354$	0	0
$355$	3585.40	0.536037
$356$	0	0
$357$	−62.8483	−0.00931733
$358$	0	0
$359$	−507.161	−0.0745597	−0.0372799	−	0.999305i	$-0.511869\pi$
$359$	−507.161	−0.0745597	−0.0372799	+	0.999305i	$0.511869\pi$
$360$	0	0
$361$	−6027.69	−0.878801
$362$	0	0
$363$	−726.888	−0.105101
$364$	0	0
$365$	−9566.07	−1.37181
$366$	0	0
$367$	−9627.62	−1.36937	−0.684683	−	0.728841i	$-0.740059\pi$
$367$	−9627.62	−1.36937	−0.684683	+	0.728841i	$0.740059\pi$
$368$	0	0
$369$	1882.41	0.265567
$370$	0	0
$371$	2006.98	0.280855
$372$	0	0
$373$	−7342.00	−1.01918	−0.509590	−	0.860417i	$-0.670203\pi$
$373$	−7342.00	−1.01918	−0.509590	+	0.860417i	$0.670203\pi$
$374$	0	0
$375$	1683.05	0.231766
$376$	0	0
$377$	78.0000	0.0106557
$378$	0	0
$379$	−11735.7	−1.59055	−0.795277	−	0.606246i	$-0.792675\pi$
$379$	−11735.7	−1.59055	−0.795277	+	0.606246i	$0.792675\pi$
$380$	0	0
$381$	4168.79	0.560561
$382$	0	0
$383$	6739.67	0.899168	0.449584	−	0.893238i	$-0.351572\pi$
$383$	6739.67	0.899168	0.449584	+	0.893238i	$0.351572\pi$
$384$	0	0
$385$	−3226.37	−0.427093
$386$	0	0
$387$	3278.92	0.430689
$388$	0	0
$389$	−13234.7	−1.72500	−0.862498	−	0.506060i	$-0.831102\pi$
$389$	−13234.7	−1.72500	−0.862498	+	0.506060i	$0.831102\pi$
$390$	0	0
$391$	−2.90637	−0.000375912 0
$392$	0	0
$393$	2105.90	0.270302
$394$	0	0
$395$	−9819.21	−1.25078
$396$	0	0
$397$	−14292.8	−1.80689	−0.903446	−	0.428701i	$-0.858971\pi$
$397$	−14292.8	−1.80689	−0.903446	+	0.428701i	$0.858971\pi$
$398$	0	0
$399$	417.989	0.0524451
$400$	0	0
$401$	11779.5	1.46693	0.733466	−	0.679726i	$-0.237901\pi$
$401$	11779.5	1.46693	0.733466	+	0.679726i	$0.237901\pi$
$402$	0	0
$403$	−3568.46	−0.441086
$404$	0	0
$405$	−1363.42	−0.167282
$406$	0	0
$407$	−3345.14	−0.407401
$408$	0	0
$409$	1039.22	0.125638	0.0628189	−	0.998025i	$-0.479991\pi$
$409$	1039.22	0.125638	0.0628189	+	0.998025i	$0.479991\pi$
$410$	0	0
$411$	3132.62	0.375963
$412$	0	0
$413$	−376.927	−0.0449089
$414$	0	0
$415$	−2737.75	−0.323834
$416$	0	0
$417$	−5805.99	−0.681824
$418$	0	0
$419$	6085.74	0.709565	0.354783	−	0.934949i	$-0.384555\pi$
$419$	6085.74	0.709565	0.354783	+	0.934949i	$0.384555\pi$
$420$	0	0
$421$	8309.91	0.961995	0.480998	−	0.876722i	$-0.340275\pi$
$421$	8309.91	0.961995	0.480998	+	0.876722i	$0.340275\pi$
$422$	0	0
$423$	2154.07	0.247599
$424$	0	0
$425$	−686.391	−0.0783409
$426$	0	0
$427$	3933.35	0.445781
$428$	0	0
$429$	−1546.93	−0.174094
$430$	0	0
$431$	−15601.0	−1.74356	−0.871778	−	0.489902i	$-0.837033\pi$
$431$	−15601.0	−1.74356	−0.871778	+	0.489902i	$0.837033\pi$
$432$	0	0
$433$	7887.07	0.875354	0.437677	−	0.899132i	$-0.355801\pi$
$433$	7887.07	0.875354	0.437677	+	0.899132i	$0.355801\pi$
$434$	0	0
$435$	−302.983	−0.0333952
$436$	0	0
$437$	19.3296	0.00211593
$438$	0	0
$439$	2105.13	0.228866	0.114433	−	0.993431i	$-0.463495\pi$
$439$	2105.13	0.228866	0.114433	+	0.993431i	$0.463495\pi$
$440$	0	0
$441$	−2876.83	−0.310639
$442$	0	0
$443$	−1387.00	−0.148754	−0.0743772	−	0.997230i	$-0.523697\pi$
$443$	−1387.00	−0.148754	−0.0743772	+	0.997230i	$0.523697\pi$
$444$	0	0
$445$	−23858.5	−2.54157
$446$	0	0
$447$	6757.12	0.714991
$448$	0	0
$449$	12563.5	1.32050	0.660252	−	0.751044i	$-0.270449\pi$
$449$	12563.5	1.32050	0.660252	+	0.751044i	$0.270449\pi$
$450$	0	0
$451$	−8296.14	−0.866187
$452$	0	0
$453$	−1878.38	−0.194822
$454$	0	0
$455$	−1057.43	−0.108952
$456$	0	0
$457$	−12331.4	−1.26222	−0.631112	−	0.775692i	$-0.717401\pi$
$457$	−12331.4	−1.26222	−0.631112	+	0.775692i	$0.717401\pi$
$458$	0	0
$459$	117.051	0.0119029
$460$	0	0
$461$	1825.26	0.184406	0.0922029	−	0.995740i	$-0.470609\pi$
$461$	1825.26	0.184406	0.0922029	+	0.995740i	$0.470609\pi$
$462$	0	0
$463$	8172.00	0.820270	0.410135	−	0.912025i	$-0.365482\pi$
$463$	8172.00	0.820270	0.410135	+	0.912025i	$0.365482\pi$
$464$	0	0
$465$	13861.3	1.38237
$466$	0	0
$467$	9959.22	0.986847	0.493424	−	0.869789i	$-0.335745\pi$
$467$	9959.22	0.986847	0.493424	+	0.869789i	$0.335745\pi$
$468$	0	0
$469$	−4148.38	−0.408432
$470$	0	0
$471$	−3307.67	−0.323587
$472$	0	0
$473$	−14450.8	−1.40476
$474$	0	0
$475$	4565.02	0.440963
$476$	0	0
$477$	−3737.87	−0.358795
$478$	0	0
$479$	−2793.32	−0.266451	−0.133226	−	0.991086i	$-0.542533\pi$
$479$	−2793.32	−0.266451	−0.133226	+	0.991086i	$0.542533\pi$
$480$	0	0
$481$	−1096.36	−0.103928
$482$	0	0
$483$	9.71909	0.000915598 0
$484$	0	0
$485$	13448.2	1.25908
$486$	0	0
$487$	−3870.17	−0.360111	−0.180056	−	0.983656i	$-0.557628\pi$
$487$	−3870.17	−0.360111	−0.180056	+	0.983656i	$0.557628\pi$
$488$	0	0
$489$	−2085.44	−0.192856
$490$	0	0
$491$	−1888.65	−0.173592	−0.0867958	−	0.996226i	$-0.527663\pi$
$491$	−1888.65	−0.173592	−0.0867958	+	0.996226i	$0.527663\pi$
$492$	0	0
$493$	26.0112	0.00237624
$494$	0	0
$495$	6008.88	0.545614
$496$	0	0
$497$	1029.33	0.0929007
$498$	0	0
$499$	−15879.0	−1.42453	−0.712266	−	0.701910i	$-0.752331\pi$
$499$	−15879.0	−1.42453	−0.712266	+	0.701910i	$0.752331\pi$
$500$	0	0
$501$	−5953.51	−0.530904
$502$	0	0
$503$	16338.7	1.44833	0.724164	−	0.689628i	$-0.242226\pi$
$503$	16338.7	1.44833	0.724164	+	0.689628i	$0.242226\pi$
$504$	0	0
$505$	27430.6	2.41712
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	−11875.7	−1.03415	−0.517076	−	0.855940i	$-0.672979\pi$
$509$	−11875.7	−1.03415	−0.517076	+	0.855940i	$0.672979\pi$
$510$	0	0
$511$	−2746.31	−0.237749
$512$	0	0
$513$	−778.475	−0.0669990
$514$	0	0
$515$	6479.24	0.554388
$516$	0	0
$517$	−9493.40	−0.807581
$518$	0	0
$519$	3302.54	0.279317
$520$	0	0
$521$	4971.92	0.418088	0.209044	−	0.977906i	$-0.432965\pi$
$521$	4971.92	0.418088	0.209044	+	0.977906i	$0.432965\pi$
$522$	0	0
$523$	1131.76	0.0946238	0.0473119	−	0.998880i	$-0.484935\pi$
$523$	1131.76	0.0946238	0.0473119	+	0.998880i	$0.484935\pi$
$524$	0	0
$525$	2295.33	0.190813
$526$	0	0
$527$	−1190.00	−0.0983630
$528$	0	0
$529$	−12166.6	−0.999963
$530$	0	0
$531$	702.000	0.0573714
$532$	0	0
$533$	−2719.03	−0.220965
$534$	0	0
$535$	5656.44	0.457102
$536$	0	0
$537$	11377.9	0.914323
$538$	0	0
$539$	12678.8	1.01320
$540$	0	0
$541$	−10514.6	−0.835593	−0.417797	−	0.908541i	$-0.637198\pi$
$541$	−10514.6	−0.835593	−0.417797	+	0.908541i	$0.637198\pi$
$542$	0	0
$543$	−5779.40	−0.456755
$544$	0	0
$545$	−13083.9	−1.02836
$546$	0	0
$547$	−21257.4	−1.66161	−0.830806	−	0.556562i	$-0.812120\pi$
$547$	−21257.4	−1.66161	−0.830806	+	0.556562i	$0.812120\pi$
$548$	0	0
$549$	−7325.60	−0.569488
$550$	0	0
$551$	−172.994	−0.0133753
$552$	0	0
$553$	−2818.99	−0.216773
$554$	0	0
$555$	4258.69	0.325714
$556$	0	0
$557$	2350.31	0.178789	0.0893947	−	0.995996i	$-0.471507\pi$
$557$	2350.31	0.178789	0.0893947	+	0.995996i	$0.471507\pi$
$558$	0	0
$559$	−4736.21	−0.358355
$560$	0	0
$561$	−515.865	−0.0388233
$562$	0	0
$563$	19787.9	1.48128	0.740639	−	0.671904i	$-0.234523\pi$
$563$	19787.9	1.48128	0.740639	+	0.671904i	$0.234523\pi$
$564$	0	0
$565$	5700.26	0.424445
$566$	0	0
$567$	−391.424	−0.0289917
$568$	0	0
$569$	−20580.0	−1.51627	−0.758136	−	0.652097i	$-0.773890\pi$
$569$	−20580.0	−1.51627	−0.758136	+	0.652097i	$0.773890\pi$
$570$	0	0
$571$	−13301.9	−0.974897	−0.487449	−	0.873152i	$-0.662072\pi$
$571$	−13301.9	−0.974897	−0.487449	+	0.873152i	$0.662072\pi$
$572$	0	0
$573$	−6145.54	−0.448052
$574$	0	0
$575$	106.146	0.00769843
$576$	0	0
$577$	−6423.54	−0.463458	−0.231729	−	0.972780i	$-0.574438\pi$
$577$	−6423.54	−0.463458	−0.231729	+	0.972780i	$0.574438\pi$
$578$	0	0
$579$	−12404.0	−0.890316
$580$	0	0
$581$	−785.979	−0.0561238
$582$	0	0
$583$	16473.5	1.17026
$584$	0	0
$585$	1969.39	0.139187
$586$	0	0
$587$	18600.5	1.30788	0.653941	−	0.756546i	$-0.273115\pi$
$587$	18600.5	1.30788	0.653941	+	0.756546i	$0.273115\pi$
$588$	0	0
$589$	7914.41	0.553663
$590$	0	0
$591$	−5570.93	−0.387745
$592$	0	0
$593$	11942.3	0.827001	0.413501	−	0.910504i	$-0.364306\pi$
$593$	11942.3	0.827001	0.413501	+	0.910504i	$0.364306\pi$
$594$	0	0
$595$	−352.629	−0.0242965
$596$	0	0
$597$	−16546.7	−1.13436
$598$	0	0
$599$	746.614	0.0509279	0.0254640	−	0.999676i	$-0.491894\pi$
$599$	746.614	0.0509279	0.0254640	+	0.999676i	$0.491894\pi$
$600$	0	0
$601$	11287.4	0.766096	0.383048	−	0.923728i	$-0.374874\pi$
$601$	11287.4	0.766096	0.383048	+	0.923728i	$0.374874\pi$
$602$	0	0
$603$	7726.07	0.521774
$604$	0	0
$605$	−4078.42	−0.274068
$606$	0	0
$607$	7090.08	0.474098	0.237049	−	0.971498i	$-0.423820\pi$
$607$	7090.08	0.474098	0.237049	+	0.971498i	$0.423820\pi$
$608$	0	0
$609$	−86.9831	−0.00578774
$610$	0	0
$611$	−3111.43	−0.206015
$612$	0	0
$613$	21842.3	1.43915	0.719577	−	0.694413i	$-0.244336\pi$
$613$	21842.3	1.43915	0.719577	+	0.694413i	$0.244336\pi$
$614$	0	0
$615$	10561.8	0.692509
$616$	0	0
$617$	7845.18	0.511888	0.255944	−	0.966692i	$-0.417614\pi$
$617$	7845.18	0.511888	0.255944	+	0.966692i	$0.417614\pi$
$618$	0	0
$619$	29667.4	1.92639	0.963193	−	0.268809i	$-0.0866302\pi$
$619$	29667.4	1.92639	0.963193	+	0.268809i	$0.0866302\pi$
$620$	0	0
$621$	−18.1011	−0.00116968
$622$	0	0
$623$	−6849.50	−0.440481
$624$	0	0
$625$	−10347.9	−0.662268
$626$	0	0
$627$	3430.89	0.218527
$628$	0	0
$629$	−365.610	−0.0231762
$630$	0	0
$631$	24321.2	1.53441	0.767204	−	0.641404i	$-0.221648\pi$
$631$	24321.2	1.53441	0.767204	+	0.641404i	$0.221648\pi$
$632$	0	0
$633$	−12107.2	−0.760217
$634$	0	0
$635$	23390.3	1.46175
$636$	0	0
$637$	4155.42	0.258468
$638$	0	0
$639$	−1917.05	−0.118681
$640$	0	0
$641$	16751.4	1.03220	0.516099	−	0.856529i	$-0.327384\pi$
$641$	16751.4	1.03220	0.516099	+	0.856529i	$0.327384\pi$
$642$	0	0
$643$	−27260.5	−1.67193	−0.835964	−	0.548784i	$-0.815091\pi$
$643$	−27260.5	−1.67193	−0.835964	+	0.548784i	$0.815091\pi$
$644$	0	0
$645$	18397.3	1.12309
$646$	0	0
$647$	−9550.09	−0.580298	−0.290149	−	0.956981i	$-0.593705\pi$
$647$	−9550.09	−0.580298	−0.290149	+	0.956981i	$0.593705\pi$
$648$	0	0
$649$	−3093.85	−0.187125
$650$	0	0
$651$	3979.44	0.239580
$652$	0	0
$653$	−1093.89	−0.0655548	−0.0327774	−	0.999463i	$-0.510435\pi$
$653$	−1093.89	−0.0655548	−0.0327774	+	0.999463i	$0.510435\pi$
$654$	0	0
$655$	11815.8	0.704856
$656$	0	0
$657$	5114.81	0.303726
$658$	0	0
$659$	−9129.47	−0.539657	−0.269828	−	0.962908i	$-0.586967\pi$
$659$	−9129.47	−0.539657	−0.269828	+	0.962908i	$0.586967\pi$
$660$	0	0
$661$	10726.5	0.631181	0.315591	−	0.948895i	$-0.397797\pi$
$661$	10726.5	0.631181	0.315591	+	0.948895i	$0.397797\pi$
$662$	0	0
$663$	−169.073	−0.00990385
$664$	0	0
$665$	2345.25	0.136759
$666$	0	0
$667$	−4.02247	−0.000233509 0
$668$	0	0
$669$	13358.2	0.771983
$670$	0	0
$671$	32285.4	1.85747
$672$	0	0
$673$	−6797.04	−0.389311	−0.194656	−	0.980872i	$-0.562359\pi$
$673$	−6797.04	−0.389311	−0.194656	+	0.980872i	$0.562359\pi$
$674$	0	0
$675$	−4274.90	−0.243764
$676$	0	0
$677$	−22167.0	−1.25841	−0.629207	−	0.777238i	$-0.716620\pi$
$677$	−22167.0	−1.25841	−0.629207	+	0.777238i	$0.716620\pi$
$678$	0	0
$679$	3860.84	0.218211
$680$	0	0
$681$	−16448.0	−0.925536
$682$	0	0
$683$	16756.8	0.938773	0.469387	−	0.882993i	$-0.344475\pi$
$683$	16756.8	0.938773	0.469387	+	0.882993i	$0.344475\pi$
$684$	0	0
$685$	17576.5	0.980385
$686$	0	0
$687$	4220.21	0.234369
$688$	0	0
$689$	5399.14	0.298535
$690$	0	0
$691$	−1807.58	−0.0995132	−0.0497566	−	0.998761i	$-0.515845\pi$
$691$	−1807.58	−0.0995132	−0.0497566	+	0.998761i	$0.515845\pi$
$692$	0	0
$693$	1725.08	0.0945607
$694$	0	0
$695$	−32576.2	−1.77797
$696$	0	0
$697$	−906.736	−0.0492756
$698$	0	0
$699$	−19260.5	−1.04220
$700$	0	0
$701$	−26699.0	−1.43853	−0.719264	−	0.694737i	$-0.755521\pi$
$701$	−26699.0	−1.43853	−0.719264	+	0.694737i	$0.755521\pi$
$702$	0	0
$703$	2431.59	0.130454
$704$	0	0
$705$	12086.0	0.645655
$706$	0	0
$707$	7875.02	0.418912
$708$	0	0
$709$	19226.1	1.01841	0.509204	−	0.860646i	$-0.329940\pi$
$709$	19226.1	1.01841	0.509204	+	0.860646i	$0.329940\pi$
$710$	0	0
$711$	5250.17	0.276929
$712$	0	0
$713$	184.026	0.00966597
$714$	0	0
$715$	−8679.50	−0.453979
$716$	0	0
$717$	14698.3	0.765577
$718$	0	0
$719$	−14372.2	−0.745470	−0.372735	−	0.927938i	$-0.621580\pi$
$719$	−14372.2	−0.745470	−0.372735	+	0.927938i	$0.621580\pi$
$720$	0	0
$721$	1860.12	0.0960811
$722$	0	0
$723$	−777.388	−0.0399880
$724$	0	0
$725$	−949.978	−0.0486638
$726$	0	0
$727$	26347.2	1.34410	0.672052	−	0.740504i	$-0.265413\pi$
$727$	26347.2	1.34410	0.672052	+	0.740504i	$0.265413\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−1579.42	−0.0799137
$732$	0	0
$733$	27597.8	1.39065	0.695326	−	0.718694i	$-0.255260\pi$
$733$	27597.8	1.39065	0.695326	+	0.718694i	$0.255260\pi$
$734$	0	0
$735$	−16141.3	−0.810043
$736$	0	0
$737$	−34050.3	−1.70185
$738$	0	0
$739$	9546.08	0.475180	0.237590	−	0.971365i	$-0.423642\pi$
$739$	9546.08	0.475180	0.237590	+	0.971365i	$0.423642\pi$
$740$	0	0
$741$	1124.46	0.0557466
$742$	0	0
$743$	−32976.0	−1.62823	−0.814114	−	0.580705i	$-0.802777\pi$
$743$	−32976.0	−1.62823	−0.814114	+	0.580705i	$0.802777\pi$
$744$	0	0
$745$	37912.8	1.86446
$746$	0	0
$747$	1463.83	0.0716985
$748$	0	0
$749$	1623.90	0.0792204
$750$	0	0
$751$	−9781.24	−0.475263	−0.237631	−	0.971355i	$-0.576371\pi$
$751$	−9781.24	−0.475263	−0.237631	+	0.971355i	$0.576371\pi$
$752$	0	0
$753$	−2654.28	−0.128456
$754$	0	0
$755$	−10539.2	−0.508029
$756$	0	0
$757$	4414.55	0.211955	0.105977	−	0.994369i	$-0.466203\pi$
$757$	4414.55	0.211955	0.105977	+	0.994369i	$0.466203\pi$
$758$	0	0
$759$	79.7753	0.00381510
$760$	0	0
$761$	−33676.2	−1.60415	−0.802077	−	0.597220i	$-0.796272\pi$
$761$	−33676.2	−1.60415	−0.802077	+	0.597220i	$0.796272\pi$
$762$	0	0
$763$	−3756.26	−0.178225
$764$	0	0
$765$	656.747	0.0310389
$766$	0	0
$767$	−1014.00	−0.0477359
$768$	0	0
$769$	−34795.0	−1.63165	−0.815827	−	0.578297i	$-0.803718\pi$
$769$	−34795.0	−1.63165	−0.815827	+	0.578297i	$0.803718\pi$
$770$	0	0
$771$	−8739.02	−0.408208
$772$	0	0
$773$	28152.1	1.30991	0.654954	−	0.755668i	$-0.272688\pi$
$773$	28152.1	1.30991	0.654954	+	0.755668i	$0.272688\pi$
$774$	0	0
$775$	43461.0	2.01441
$776$	0	0
$777$	1222.62	0.0564496
$778$	0	0
$779$	6030.48	0.277361
$780$	0	0
$781$	8448.82	0.387097
$782$	0	0
$783$	162.000	0.00739388
$784$	0	0
$785$	−18558.7	−0.843807
$786$	0	0
$787$	2659.15	0.120443	0.0602215	−	0.998185i	$-0.480819\pi$
$787$	2659.15	0.120443	0.0602215	+	0.998185i	$0.480819\pi$
$788$	0	0
$789$	381.989	0.0172359
$790$	0	0
$791$	1636.48	0.0735608
$792$	0	0
$793$	10581.4	0.473842
$794$	0	0
$795$	−20972.4	−0.935616
$796$	0	0
$797$	33193.5	1.47525	0.737625	−	0.675210i	$-0.235947\pi$
$797$	33193.5	1.47525	0.737625	+	0.675210i	$0.235947\pi$
$798$	0	0
$799$	−1037.59	−0.0459416
$800$	0	0
$801$	12756.7	0.562717
$802$	0	0
$803$	−22542.0	−0.990648
$804$	0	0
$805$	54.5319	0.00238757
$806$	0	0
$807$	−24532.5	−1.07012
$808$	0	0
$809$	9198.73	0.399765	0.199883	−	0.979820i	$-0.435944\pi$
$809$	9198.73	0.399765	0.199883	+	0.979820i	$0.435944\pi$
$810$	0	0
$811$	2435.11	0.105436	0.0527179	−	0.998609i	$-0.483212\pi$
$811$	2435.11	0.105436	0.0527179	+	0.998609i	$0.483212\pi$
$812$	0	0
$813$	12341.5	0.532393
$814$	0	0
$815$	−11701.0	−0.502904
$816$	0	0
$817$	10504.3	0.449817
$818$	0	0
$819$	565.390	0.0241225
$820$	0	0
$821$	17767.2	0.755275	0.377638	−	0.925954i	$-0.376737\pi$
$821$	17767.2	0.755275	0.377638	+	0.925954i	$0.376737\pi$
$822$	0	0
$823$	−15112.0	−0.640060	−0.320030	−	0.947407i	$-0.603693\pi$
$823$	−15112.0	−0.640060	−0.320030	+	0.947407i	$0.603693\pi$
$824$	0	0
$825$	18840.3	0.795074
$826$	0	0
$827$	32606.2	1.37102	0.685508	−	0.728065i	$-0.259580\pi$
$827$	32606.2	1.37102	0.685508	+	0.728065i	$0.259580\pi$
$828$	0	0
$829$	−26701.9	−1.11869	−0.559347	−	0.828934i	$-0.688948\pi$
$829$	−26701.9	−1.11869	−0.559347	+	0.828934i	$0.688948\pi$
$830$	0	0
$831$	−1762.63	−0.0735799
$832$	0	0
$833$	1385.74	0.0576387
$834$	0	0
$835$	−33403.9	−1.38442
$836$	0	0
$837$	−7411.42	−0.306065
$838$	0	0
$839$	−20747.5	−0.853736	−0.426868	−	0.904314i	$-0.640383\pi$
$839$	−20747.5	−0.853736	−0.426868	+	0.904314i	$0.640383\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	0	0
$843$	−11775.4	−0.481099
$844$	0	0
$845$	−2844.68	−0.115810
$846$	0	0
$847$	−1170.87	−0.0474989
$848$	0	0
$849$	−19383.8	−0.783568
$850$	0	0
$851$	56.5393	0.00227749
$852$	0	0
$853$	4041.50	0.162225	0.0811127	−	0.996705i	$-0.474153\pi$
$853$	4041.50	0.162225	0.0811127	+	0.996705i	$0.474153\pi$
$854$	0	0
$855$	−4367.87	−0.174711
$856$	0	0
$857$	−32067.4	−1.27818	−0.639091	−	0.769131i	$-0.720689\pi$
$857$	−32067.4	−1.27818	−0.639091	+	0.769131i	$0.720689\pi$
$858$	0	0
$859$	−3739.80	−0.148545	−0.0742726	−	0.997238i	$-0.523664\pi$
$859$	−3739.80	−0.148545	−0.0742726	+	0.997238i	$0.523664\pi$
$860$	0	0
$861$	3032.18	0.120019
$862$	0	0
$863$	38258.1	1.50906	0.754531	−	0.656265i	$-0.227865\pi$
$863$	38258.1	1.50906	0.754531	+	0.656265i	$0.227865\pi$
$864$	0	0
$865$	18529.9	0.728365
$866$	0	0
$867$	14682.6	0.575142
$868$	0	0
$869$	−23138.5	−0.903246
$870$	0	0
$871$	−11159.9	−0.434142
$872$	0	0
$873$	−7190.54	−0.278766
$874$	0	0
$875$	2711.06	0.104743
$876$	0	0
$877$	−21528.7	−0.828932	−0.414466	−	0.910065i	$-0.636032\pi$
$877$	−21528.7	−0.828932	−0.414466	+	0.910065i	$0.636032\pi$
$878$	0	0
$879$	−28811.5	−1.10556
$880$	0	0
$881$	25995.3	0.994101	0.497051	−	0.867722i	$-0.334416\pi$
$881$	25995.3	0.994101	0.497051	+	0.867722i	$0.334416\pi$
$882$	0	0
$883$	9586.18	0.365346	0.182673	−	0.983174i	$-0.441525\pi$
$883$	9586.18	0.365346	0.182673	+	0.983174i	$0.441525\pi$
$884$	0	0
$885$	3938.78	0.149605
$886$	0	0
$887$	20879.1	0.790361	0.395180	−	0.918604i	$-0.370682\pi$
$887$	20879.1	0.790361	0.395180	+	0.918604i	$0.370682\pi$
$888$	0	0
$889$	6715.09	0.253337
$890$	0	0
$891$	−3212.85	−0.120802
$892$	0	0
$893$	6900.77	0.258595
$894$	0	0
$895$	63839.0	2.38425
$896$	0	0
$897$	26.1461	0.000973235 0
$898$	0	0
$899$	−1646.98	−0.0611012
$900$	0	0
$901$	1800.49	0.0665739
$902$	0	0
$903$	5281.67	0.194644
$904$	0	0
$905$	−32427.1	−1.19106
$906$	0	0
$907$	1094.00	0.0400503	0.0200251	−	0.999799i	$-0.493625\pi$
$907$	1094.00	0.0400503	0.0200251	+	0.999799i	$0.493625\pi$
$908$	0	0
$909$	−14666.7	−0.535163
$910$	0	0
$911$	−29816.7	−1.08438	−0.542190	−	0.840256i	$-0.682405\pi$
$911$	−29816.7	−1.08438	−0.542190	+	0.840256i	$0.682405\pi$
$912$	0	0
$913$	−6451.40	−0.233856
$914$	0	0
$915$	−41102.4	−1.48503
$916$	0	0
$917$	3392.18	0.122159
$918$	0	0
$919$	−40744.8	−1.46251	−0.731255	−	0.682104i	$-0.761065\pi$
$919$	−40744.8	−1.46251	−0.731255	+	0.682104i	$0.761065\pi$
$920$	0	0
$921$	−15939.6	−0.570281
$922$	0	0
$923$	2769.07	0.0987488
$924$	0	0
$925$	13352.8	0.474634
$926$	0	0
$927$	−3464.34	−0.122744
$928$	0	0
$929$	−40223.7	−1.42056	−0.710278	−	0.703921i	$-0.751431\pi$
$929$	−40223.7	−1.42056	−0.710278	+	0.703921i	$0.751431\pi$
$930$	0	0
$931$	−9216.22	−0.324435
$932$	0	0
$933$	−19988.6	−0.701392
$934$	0	0
$935$	−2894.42	−0.101238
$936$	0	0
$937$	−511.273	−0.0178256	−0.00891279	−	0.999960i	$-0.502837\pi$
$937$	−511.273	−0.0178256	−0.00891279	+	0.999960i	$0.502837\pi$
$938$	0	0
$939$	−4670.11	−0.162304
$940$	0	0
$941$	24696.7	0.855569	0.427785	−	0.903881i	$-0.359294\pi$
$941$	24696.7	0.855569	0.427785	+	0.903881i	$0.359294\pi$
$942$	0	0
$943$	140.221	0.00484223
$944$	0	0
$945$	−2196.20	−0.0756005
$946$	0	0
$947$	−11177.7	−0.383555	−0.191778	−	0.981438i	$-0.561425\pi$
$947$	−11177.7	−0.383555	−0.191778	+	0.981438i	$0.561425\pi$
$948$	0	0
$949$	−7388.07	−0.252715
$950$	0	0
$951$	1987.99	0.0677866
$952$	0	0
$953$	31891.5	1.08402	0.542008	−	0.840374i	$-0.317664\pi$
$953$	31891.5	1.08402	0.542008	+	0.840374i	$0.317664\pi$
$954$	0	0
$955$	−34481.4	−1.16837
$956$	0	0
$957$	−713.966	−0.0241163
$958$	0	0
$959$	5046.02	0.169911
$960$	0	0
$961$	45557.7	1.52924
$962$	0	0
$963$	−3024.40	−0.101205
$964$	0	0
$965$	−69596.4	−2.32165
$966$	0	0
$967$	−46487.2	−1.54594	−0.772971	−	0.634441i	$-0.781230\pi$
$967$	−46487.2	−1.54594	−0.772971	+	0.634441i	$0.781230\pi$
$968$	0	0
$969$	374.983	0.0124316
$970$	0	0
$971$	45959.6	1.51897	0.759483	−	0.650528i	$-0.225452\pi$
$971$	45959.6	1.51897	0.759483	+	0.650528i	$0.225452\pi$
$972$	0	0
$973$	−9352.28	−0.308140
$974$	0	0
$975$	6174.85	0.202824
$976$	0	0
$977$	3859.76	0.126392	0.0631959	−	0.998001i	$-0.479871\pi$
$977$	3859.76	0.126392	0.0631959	+	0.998001i	$0.479871\pi$
$978$	0	0
$979$	−56221.4	−1.83539
$980$	0	0
$981$	6995.76	0.227684
$982$	0	0
$983$	−12285.9	−0.398637	−0.199318	−	0.979935i	$-0.563873\pi$
$983$	−12285.9	−0.398637	−0.199318	+	0.979935i	$0.563873\pi$
$984$	0	0
$985$	−31257.4	−1.01111
$986$	0	0
$987$	3469.77	0.111899
$988$	0	0
$989$	244.247	0.00785299
$990$	0	0
$991$	5559.34	0.178202	0.0891011	−	0.996023i	$-0.471601\pi$
$991$	5559.34	0.178202	0.0891011	+	0.996023i	$0.471601\pi$
$992$	0	0
$993$	−4136.49	−0.132193
$994$	0	0
$995$	−92840.2	−2.95802
$996$	0	0
$997$	35491.1	1.12740	0.563698	−	0.825981i	$-0.309378\pi$
$997$	35491.1	1.12740	0.563698	+	0.825981i	$0.309378\pi$
$998$	0	0
$999$	−2277.05	−0.0721148

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	312.4.a.b.1.1	✓	2
3.2	odd	2		936.4.a.h.1.2		2
4.3	odd	2		624.4.a.n.1.1		2
8.3	odd	2		2496.4.a.x.1.2		2
8.5	even	2		2496.4.a.bi.1.2		2
12.11	even	2		1872.4.a.bd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.b.1.1	✓	2	1.1	even	1	trivial
624.4.a.n.1.1		2	4.3	odd	2
936.4.a.h.1.2		2	3.2	odd	2
1872.4.a.bd.1.2		2	12.11	even	2
2496.4.a.x.1.2		2	8.3	odd	2
2496.4.a.bi.1.2		2	8.5	even	2

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:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	312.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -16.8324 q^{5} -4.83240 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -16.8324 q^{5} -4.83240 q^{7} +9.00000 q^{9} -39.6648 q^{11} -13.0000 q^{13} +50.4972 q^{15} -4.33521 q^{17} +28.8324 q^{19} +14.4972 q^{21} +0.670412 q^{23} +158.330 q^{25} -27.0000 q^{27} -6.00000 q^{29} +274.497 q^{31} +118.994 q^{33} +81.3408 q^{35} +84.3352 q^{37} +39.0000 q^{39} +209.156 q^{41} +364.324 q^{43} -151.492 q^{45} +239.341 q^{47} -319.648 q^{49} +13.0056 q^{51} -415.318 q^{53} +667.654 q^{55} -86.4972 q^{57} +78.0000 q^{59} -813.955 q^{61} -43.4916 q^{63} +218.821 q^{65} +858.452 q^{67} -2.01124 q^{69} -213.006 q^{71} +568.313 q^{73} -474.989 q^{75} +191.676 q^{77} +583.352 q^{79} +81.0000 q^{81} +162.648 q^{83} +72.9719 q^{85} +18.0000 q^{87} +1417.41 q^{89} +62.8212 q^{91} -823.492 q^{93} -485.318 q^{95} -798.949 q^{97} -356.983 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 4 q^{5} + 20 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 - 4 * q^5 + 20 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} - 4 q^{5} + 20 q^{7} + 18 q^{9} - 20 q^{11} - 26 q^{13} + 12 q^{15} - 68 q^{17} + 28 q^{19} - 60 q^{21} + 120 q^{23} + 198 q^{25} - 54 q^{27} - 12 q^{29} + 460 q^{31} + 60 q^{33} + 400 q^{35} + 228 q^{37} + 78 q^{39} + 92 q^{41} + 432 q^{43} - 36 q^{45} + 716 q^{47} - 46 q^{49} + 204 q^{51} - 356 q^{53} + 920 q^{55} - 84 q^{57} + 156 q^{59} - 204 q^{61} + 180 q^{63} + 52 q^{65} + 204 q^{67} - 360 q^{69} - 604 q^{71} + 484 q^{73} - 594 q^{75} + 680 q^{77} + 1760 q^{79} + 162 q^{81} - 268 q^{83} - 744 q^{85} + 36 q^{87} + 76 q^{89} - 260 q^{91} - 1380 q^{93} - 496 q^{95} + 4 q^{97} - 180 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 4 * q^5 + 20 * q^7 + 18 * q^9 - 20 * q^11 - 26 * q^13 + 12 * q^15 - 68 * q^17 + 28 * q^19 - 60 * q^21 + 120 * q^23 + 198 * q^25 - 54 * q^27 - 12 * q^29 + 460 * q^31 + 60 * q^33 + 400 * q^35 + 228 * q^37 + 78 * q^39 + 92 * q^41 + 432 * q^43 - 36 * q^45 + 716 * q^47 - 46 * q^49 + 204 * q^51 - 356 * q^53 + 920 * q^55 - 84 * q^57 + 156 * q^59 - 204 * q^61 + 180 * q^63 + 52 * q^65 + 204 * q^67 - 360 * q^69 - 604 * q^71 + 484 * q^73 - 594 * q^75 + 680 * q^77 + 1760 * q^79 + 162 * q^81 - 268 * q^83 - 744 * q^85 + 36 * q^87 + 76 * q^89 - 260 * q^91 - 1380 * q^93 - 496 * q^95 + 4 * q^97 - 180 * q^99

Introduction

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