LMFDB - Embedded newform 3100.3.d.e.1301.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3100,3,Mod(1301,3100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3100, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("3100.1301");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3100 = 2^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3100.d (of order $2$, degree $1$, 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:	$84.4688819517$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 114 x^{18} + 5280 x^{16} + 128422 x^{14} + 1776819 x^{12} + 14249420 x^{10} + 65297060 x^{8} + \cdots + 20793600 $ x^20 + 114x^18 + 5280x^16 + 128422x^14 + 1776819x^12 + 14249420x^10 + 65297060x^8 + 162416640x^6 + 199076000x^4 + 107961600*x^2 + 20793600
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{23}\cdot 3 $
Twist minimal:	no (minimal twist has level 620)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1301.14
Root			$2.04380i$ of defining polynomial
Character	$\chi$	$=$	3100.1301
Dual form			3100.3.d.e.1301.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.04380i q^{3} +1.40634 q^{7} +4.82290 q^{9} +O(q^{10})$	$q+2.04380i q^{3} +1.40634 q^{7} +4.82290 q^{9} -10.5279i q^{11} +6.23693i q^{13} -1.99696i q^{17} -17.8889 q^{19} +2.87428i q^{21} -26.5067i q^{23} +28.2512i q^{27} -39.1096i q^{29} +(-29.7303 + 8.78105i) q^{31} +21.5169 q^{33} +43.7982i q^{37} -12.7470 q^{39} -36.1724 q^{41} +11.2206i q^{43} +35.1874 q^{47} -47.0222 q^{49} +4.08137 q^{51} -19.7784i q^{53} -36.5612i q^{57} -64.5921 q^{59} +49.0630i q^{61} +6.78265 q^{63} -44.7351 q^{67} +54.1743 q^{69} -94.7163 q^{71} +96.5778i q^{73} -14.8059i q^{77} +79.9248i q^{79} -14.3336 q^{81} +19.4022i q^{83} +79.9320 q^{87} -54.2141i q^{89} +8.77126i q^{91} +(-17.9467 - 60.7627i) q^{93} +93.5642 q^{97} -50.7751i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 12 q^{7} - 48 q^{9}+O(q^{10}) $ 20 * q - 12 * q^7 - 48 * q^9	$ 20 q - 12 q^{7} - 48 q^{9} + 60 q^{19} + 8 q^{31} - 68 q^{33} + 28 q^{39} - 80 q^{41} + 48 q^{47} + 84 q^{49} + 344 q^{51} + 160 q^{59} + 232 q^{63} + 180 q^{67} - 140 q^{69} - 108 q^{71} + 336 q^{81} + 236 q^{87} + 332 q^{93} + 112 q^{97}+O(q^{100}) $ 20 * q - 12 * q^7 - 48 * q^9 + 60 * q^19 + 8 * q^31 - 68 * q^33 + 28 * q^39 - 80 * q^41 + 48 * q^47 + 84 * q^49 + 344 * q^51 + 160 * q^59 + 232 * q^63 + 180 * q^67 - 140 * q^69 - 108 * q^71 + 336 * q^81 + 236 * q^87 + 332 * q^93 + 112 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3100\mathbb{Z}\right)^\times$.

$n$	$1551$	$1801$	$2977$
$\chi(n)$	$1$	$-1$	$1$

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
$3$			2.04380i			0.681265i	0.940196	+	0.340633i	$0.110641\pi$
$3$			2.04380i			0.681265i	−0.940196	+	0.340633i	$0.889359\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.40634			0.200906			0.100453	−	0.994942i	$-0.467971\pi$
$7$	1.40634			0.200906			0.100453	+	0.994942i	$0.467971\pi$
$8$		0			0
$9$	4.82290			0.535878
$10$		0			0
$11$		−	10.5279i		−	0.957085i	−0.878064	−	0.478542i	$-0.841165\pi$
$11$		−	10.5279i		−	0.957085i	0.878064	−	0.478542i	$-0.158835\pi$
$12$		0			0
$13$			6.23693i			0.479764i	0.970802	+	0.239882i	$0.0771088\pi$
$13$			6.23693i			0.479764i	−0.970802	+	0.239882i	$0.922891\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	1.99696i		−	0.117468i	−0.998274	−	0.0587340i	$-0.981294\pi$
$17$		−	1.99696i		−	0.117468i	0.998274	−	0.0587340i	$-0.0187064\pi$
$18$		0			0
$19$	−17.8889			−0.941521			−0.470760	−	0.882261i	$-0.656020\pi$
$19$	−17.8889			−0.941521			−0.470760	+	0.882261i	$0.656020\pi$
$20$		0			0
$21$			2.87428i			0.136870i
$22$		0			0
$23$		−	26.5067i		−	1.15247i	−0.817286	−	0.576233i	$-0.804522\pi$
$23$		−	26.5067i		−	1.15247i	0.817286	−	0.576233i	$-0.195478\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			28.2512i			1.04634i
$28$		0			0
$29$		−	39.1096i		−	1.34861i	−0.738454	−	0.674304i	$-0.764444\pi$
$29$		−	39.1096i		−	1.34861i	0.738454	−	0.674304i	$-0.235556\pi$
$30$		0			0
$31$	−29.7303	+	8.78105i	−0.959043	+	0.283260i
$31$	−29.7303	+	8.78105i	−0.959043	+	0.283260i
$32$		0			0
$33$	21.5169			0.652029
$34$		0			0
$35$		0			0
$36$		0			0
$37$			43.7982i			1.18374i	0.806035	+	0.591868i	$0.201609\pi$
$37$			43.7982i			1.18374i	−0.806035	+	0.591868i	$0.798391\pi$
$38$		0			0
$39$	−12.7470			−0.326847
$40$		0			0
$41$	−36.1724			−0.882255			−0.441127	−	0.897444i	$-0.645421\pi$
$41$	−36.1724			−0.882255			−0.441127	+	0.897444i	$0.645421\pi$
$42$		0			0
$43$			11.2206i			0.260945i	0.991452	+	0.130473i	$0.0416495\pi$
$43$			11.2206i			0.260945i	−0.991452	+	0.130473i	$0.958351\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	35.1874			0.748668			0.374334	−	0.927294i	$-0.377871\pi$
$47$	35.1874			0.748668			0.374334	+	0.927294i	$0.377871\pi$
$48$		0			0
$49$	−47.0222			−0.959637
$50$		0			0
$51$	4.08137			0.0800269
$52$		0			0
$53$		−	19.7784i		−	0.373177i	−0.982438	−	0.186588i	$-0.940257\pi$
$53$		−	19.7784i		−	0.373177i	0.982438	−	0.186588i	$-0.0597431\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	36.5612i		−	0.641425i
$58$		0			0
$59$	−64.5921			−1.09478			−0.547391	−	0.836877i	$-0.684379\pi$
$59$	−64.5921			−1.09478			−0.547391	+	0.836877i	$0.684379\pi$
$60$		0			0
$61$			49.0630i			0.804311i	0.915571	+	0.402156i	$0.131739\pi$
$61$			49.0630i			0.804311i	−0.915571	+	0.402156i	$0.868261\pi$
$62$		0			0
$63$	6.78265			0.107661
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−44.7351			−0.667688			−0.333844	−	0.942628i	$-0.608346\pi$
$67$	−44.7351			−0.667688			−0.333844	+	0.942628i	$0.608346\pi$
$68$		0			0
$69$	54.1743			0.785135
$70$		0			0
$71$	−94.7163			−1.33403			−0.667016	−	0.745043i	$-0.732429\pi$
$71$	−94.7163			−1.33403			−0.667016	+	0.745043i	$0.732429\pi$
$72$		0			0
$73$			96.5778i			1.32298i	0.749953	+	0.661492i	$0.230076\pi$
$73$			96.5778i			1.32298i	−0.749953	+	0.661492i	$0.769924\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	14.8059i		−	0.192284i
$78$		0			0
$79$			79.9248i			1.01171i	0.862620	+	0.505853i	$0.168822\pi$
$79$			79.9248i			1.01171i	−0.862620	+	0.505853i	$0.831178\pi$
$80$		0			0
$81$	−14.3336			−0.176958
$82$		0			0
$83$			19.4022i			0.233761i	0.993146	+	0.116880i	$0.0372894\pi$
$83$			19.4022i			0.233761i	−0.993146	+	0.116880i	$0.962711\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	79.9320			0.918759
$88$		0			0
$89$		−	54.2141i		−	0.609147i	−0.952489	−	0.304573i	$-0.901486\pi$
$89$		−	54.2141i		−	0.609147i	0.952489	−	0.304573i	$-0.0985139\pi$
$90$		0			0
$91$			8.77126i			0.0963875i
$92$		0			0
$93$	−17.9467	−	60.7627i	−0.192975	−	0.653363i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	93.5642			0.964580			0.482290	−	0.876012i	$-0.339805\pi$
$97$	93.5642			0.964580			0.482290	+	0.876012i	$0.339805\pi$
$98$		0			0
$99$		−	50.7751i		−	0.512880i
$100$		0			0
$101$	8.04238			0.0796275			0.0398137	−	0.999207i	$-0.487324\pi$
$101$	8.04238			0.0796275			0.0398137	+	0.999207i	$0.487324\pi$
$102$		0			0
$103$	−140.901			−1.36797			−0.683983	−	0.729498i	$-0.739754\pi$
$103$	−140.901			−1.36797			−0.683983	+	0.729498i	$0.739754\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	79.4295			0.742332			0.371166	−	0.928567i	$-0.378958\pi$
$107$	79.4295			0.742332			0.371166	+	0.928567i	$0.378958\pi$
$108$		0			0
$109$	79.2576			0.727134			0.363567	−	0.931568i	$-0.381559\pi$
$109$	79.2576			0.727134			0.363567	+	0.931568i	$0.381559\pi$
$110$		0			0
$111$	−89.5146			−0.806438
$112$		0			0
$113$	−133.621			−1.18249			−0.591244	−	0.806493i	$-0.701363\pi$
$113$	−133.621			−1.18249			−0.591244	+	0.806493i	$0.701363\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			30.0801i			0.257095i
$118$		0			0
$119$		−	2.80841i		−	0.0236000i
$120$		0			0
$121$	10.1626			0.0839888
$122$		0			0
$123$		−	73.9291i		−	0.601050i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			218.421i			1.71985i	0.510421	+	0.859925i	$0.329490\pi$
$127$			218.421i			1.71985i	−0.510421	+	0.859925i	$0.670510\pi$
$128$		0			0
$129$	−22.9327			−0.177773
$130$		0			0
$131$	−108.006			−0.824473			−0.412236	−	0.911077i	$-0.635252\pi$
$131$	−108.006			−0.824473			−0.412236	+	0.911077i	$0.635252\pi$
$132$		0			0
$133$	−25.1579			−0.189157
$134$		0			0
$135$		0			0
$136$		0			0
$137$			112.787i			0.823263i	0.911350	+	0.411632i	$0.135041\pi$
$137$			112.787i			0.823263i	−0.911350	+	0.411632i	$0.864959\pi$
$138$		0			0
$139$			22.4992i			0.161865i	0.996720	+	0.0809323i	$0.0257897\pi$
$139$			22.4992i			0.161865i	−0.996720	+	0.0809323i	$0.974210\pi$
$140$		0			0
$141$			71.9158i			0.510041i
$142$		0			0
$143$	65.6620			0.459175
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	96.1038i		−	0.653767i
$148$		0			0
$149$	40.1475			0.269446			0.134723	−	0.990883i	$-0.456985\pi$
$149$	40.1475			0.269446			0.134723	+	0.990883i	$0.456985\pi$
$150$		0			0
$151$			146.691i			0.971464i	0.874108	+	0.485732i	$0.161447\pi$
$151$			146.691i			0.971464i	−0.874108	+	0.485732i	$0.838553\pi$
$152$		0			0
$153$		−	9.63112i		−	0.0629485i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−225.820			−1.43834			−0.719171	−	0.694834i	$-0.755478\pi$
$157$	−225.820			−1.43834			−0.719171	+	0.694834i	$0.755478\pi$
$158$		0			0
$159$	40.4230			0.254233
$160$		0			0
$161$		−	37.2775i		−	0.231537i
$162$		0			0
$163$	−24.2440			−0.148736			−0.0743680	−	0.997231i	$-0.523694\pi$
$163$	−24.2440			−0.148736			−0.0743680	+	0.997231i	$0.523694\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			135.552i			0.811689i	0.913942	+	0.405844i	$0.133022\pi$
$167$			135.552i			0.811689i	−0.913942	+	0.405844i	$0.866978\pi$
$168$		0			0
$169$	130.101			0.769826
$170$		0			0
$171$	−86.2763			−0.504540
$172$		0			0
$173$	−263.109			−1.52086			−0.760432	−	0.649418i	$-0.775013\pi$
$173$	−263.109			−1.52086			−0.760432	+	0.649418i	$0.775013\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	132.013i		−	0.745837i
$178$		0			0
$179$		−	270.900i		−	1.51341i	−0.653759	−	0.756703i	$-0.726809\pi$
$179$		−	270.900i		−	1.51341i	0.653759	−	0.756703i	$-0.273191\pi$
$180$		0			0
$181$		−	257.743i		−	1.42399i	−0.702183	−	0.711997i	$-0.747791\pi$
$181$		−	257.743i		−	1.42399i	0.702183	−	0.711997i	$-0.252209\pi$
$182$		0			0
$183$	−100.275			−0.547949
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−21.0238			−0.112427
$188$		0			0
$189$			39.7308i			0.210216i
$190$		0			0
$191$	56.0594			0.293505			0.146752	−	0.989173i	$-0.453118\pi$
$191$	56.0594			0.293505			0.146752	+	0.989173i	$0.453118\pi$
$192$		0			0
$193$	−245.278			−1.27087			−0.635434	−	0.772155i	$-0.719179\pi$
$193$	−245.278			−1.27087			−0.635434	+	0.772155i	$0.719179\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	70.7287i		−	0.359029i	−0.983755	−	0.179514i	$-0.942547\pi$
$197$		−	70.7287i		−	0.359029i	0.983755	−	0.179514i	$-0.0574527\pi$
$198$		0			0
$199$			36.1743i			0.181781i	0.995861	+	0.0908903i	$0.0289712\pi$
$199$			36.1743i			0.181781i	−0.995861	+	0.0908903i	$0.971029\pi$
$200$		0			0
$201$		−	91.4294i		−	0.454873i
$202$		0			0
$203$		−	55.0015i		−	0.270943i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	127.839i		−	0.617580i
$208$		0			0
$209$			188.333i			0.901115i
$210$		0			0
$211$	−126.739			−0.600657			−0.300329	−	0.953836i	$-0.597096\pi$
$211$	−126.739			−0.600657			−0.300329	+	0.953836i	$0.597096\pi$
$212$		0			0
$213$		−	193.581i		−	0.908830i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−41.8110	+	12.3492i	−0.192678	+	0.0569086i
$218$		0			0
$219$	−197.385			−0.901303
$220$		0			0
$221$	12.4549			0.0563570
$222$		0			0
$223$		−	357.348i		−	1.60246i	−0.598360	−	0.801228i	$-0.704181\pi$
$223$		−	357.348i		−	1.60246i	0.598360	−	0.801228i	$-0.295819\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	54.4328			0.239792			0.119896	−	0.992786i	$-0.461744\pi$
$227$	54.4328			0.239792			0.119896	+	0.992786i	$0.461744\pi$
$228$		0			0
$229$			299.419i			1.30751i	0.756708	+	0.653753i	$0.226806\pi$
$229$			299.419i			1.30751i	−0.756708	+	0.653753i	$0.773194\pi$
$230$		0			0
$231$	30.2602			0.130996
$232$		0			0
$233$	−121.987			−0.523550			−0.261775	−	0.965129i	$-0.584308\pi$
$233$	−121.987			−0.523550			−0.261775	+	0.965129i	$0.584308\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−163.350			−0.689240
$238$		0			0
$239$		−	370.390i		−	1.54975i	−0.632116	−	0.774874i	$-0.717813\pi$
$239$		−	370.390i		−	1.54975i	0.632116	−	0.774874i	$-0.282187\pi$
$240$		0			0
$241$			159.695i			0.662635i	0.943519	+	0.331317i	$0.107493\pi$
$241$			159.695i			0.662635i	−0.943519	+	0.331317i	$0.892507\pi$
$242$		0			0
$243$			224.966i			0.925785i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	111.572i		−	0.451708i
$248$		0			0
$249$	−39.6540			−0.159253
$250$		0			0
$251$		−	173.438i		−	0.690988i	−0.938421	−	0.345494i	$-0.887711\pi$
$251$		−	173.438i		−	0.690988i	0.938421	−	0.345494i	$-0.112289\pi$
$252$		0			0
$253$	−279.061			−1.10301
$254$		0			0
$255$		0			0
$256$		0			0
$257$	202.280			0.787082			0.393541	−	0.919307i	$-0.371250\pi$
$257$	202.280			0.787082			0.393541	+	0.919307i	$0.371250\pi$
$258$		0			0
$259$			61.5953i			0.237820i
$260$		0			0
$261$		−	188.622i		−	0.722688i
$262$		0			0
$263$		−	187.288i		−	0.712120i	−0.934463	−	0.356060i	$-0.884120\pi$
$263$		−	187.288i		−	0.712120i	0.934463	−	0.356060i	$-0.115880\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	110.802			0.414990
$268$		0			0
$269$			466.621i			1.73465i	0.497743	+	0.867325i	$0.334162\pi$
$269$			466.621i			1.73465i	−0.497743	+	0.867325i	$0.665838\pi$
$270$		0			0
$271$		−	307.579i		−	1.13498i	−0.823381	−	0.567490i	$-0.807915\pi$
$271$		−	307.579i		−	1.13498i	0.823381	−	0.567490i	$-0.192085\pi$
$272$		0			0
$273$	−17.9267			−0.0656654
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	231.261i		−	0.834876i	−0.908705	−	0.417438i	$-0.862928\pi$
$277$		−	231.261i		−	0.834876i	0.908705	−	0.417438i	$-0.137072\pi$
$278$		0			0
$279$	−143.386	+	42.3501i	−0.513930	+	0.151793i
$280$		0			0
$281$	−293.826			−1.04564			−0.522822	−	0.852442i	$-0.675121\pi$
$281$	−293.826			−1.04564			−0.522822	+	0.852442i	$0.675121\pi$
$282$		0			0
$283$	−240.726			−0.850623			−0.425311	−	0.905047i	$-0.639835\pi$
$283$	−240.726			−0.850623			−0.425311	+	0.905047i	$0.639835\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−50.8708			−0.177250
$288$		0			0
$289$	285.012			0.986201
$290$		0			0
$291$			191.226i			0.657135i
$292$		0			0
$293$	−322.468			−1.10057			−0.550287	−	0.834975i	$-0.685482\pi$
$293$	−322.468			−1.10057			−0.550287	+	0.834975i	$0.685482\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	297.427			1.00144
$298$		0			0
$299$	165.321			0.552911
$300$		0			0
$301$			15.7801i			0.0524255i
$302$		0			0
$303$			16.4370i			0.0542474i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−154.337			−0.502727			−0.251364	−	0.967893i	$-0.580879\pi$
$307$	−154.337			−0.502727			−0.251364	+	0.967893i	$0.580879\pi$
$308$		0			0
$309$		−	287.972i		−	0.931948i
$310$		0			0
$311$	−423.505			−1.36175			−0.680876	−	0.732399i	$-0.738401\pi$
$311$	−423.505			−1.36175			−0.680876	+	0.732399i	$0.738401\pi$
$312$		0			0
$313$		−	144.196i		−	0.460690i	−0.973109	−	0.230345i	$-0.926015\pi$
$313$		−	144.196i		−	0.460690i	0.973109	−	0.230345i	$-0.0739855\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	434.323			1.37010			0.685051	−	0.728495i	$-0.259780\pi$
$317$	434.323			1.37010			0.685051	+	0.728495i	$0.259780\pi$
$318$		0			0
$319$	−411.743			−1.29073
$320$		0			0
$321$			162.338i			0.505725i
$322$		0			0
$323$			35.7234i			0.110599i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			161.986i			0.495371i
$328$		0			0
$329$	49.4855			0.150412
$330$		0			0
$331$		−	630.245i		−	1.90406i	−0.305997	−	0.952032i	$-0.598990\pi$
$331$		−	630.245i		−	1.90406i	0.305997	−	0.952032i	$-0.401010\pi$
$332$		0			0
$333$			211.234i			0.634337i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			480.373i			1.42544i	0.701449	+	0.712719i	$0.252537\pi$
$337$			480.373i			1.42544i	−0.701449	+	0.712719i	$0.747463\pi$
$338$		0			0
$339$		−	273.094i		−	0.805588i
$340$		0			0
$341$	92.4463	+	312.999i	0.271104	+	0.917886i
$342$		0			0
$343$	−135.040			−0.393703
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	584.757i		−	1.68518i	−0.538558	−	0.842589i	$-0.681031\pi$
$347$		−	584.757i		−	1.68518i	0.538558	−	0.842589i	$-0.318969\pi$
$348$		0			0
$349$	−132.183			−0.378748			−0.189374	−	0.981905i	$-0.560646\pi$
$349$	−132.183			−0.378748			−0.189374	+	0.981905i	$0.560646\pi$
$350$		0			0
$351$	−176.201			−0.501996
$352$		0			0
$353$			155.423i			0.440291i	0.975467	+	0.220145i	$0.0706532\pi$
$353$			155.423i			0.440291i	−0.975467	+	0.220145i	$0.929347\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	5.73981			0.0160779
$358$		0			0
$359$	85.6377			0.238545			0.119273	−	0.992862i	$-0.461944\pi$
$359$	85.6377			0.238545			0.119273	+	0.992862i	$0.461944\pi$
$360$		0			0
$361$	−40.9875			−0.113539
$362$		0			0
$363$			20.7704i			0.0572186i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	184.732i		−	0.503356i	−0.967811	−	0.251678i	$-0.919018\pi$
$367$		−	184.732i		−	0.503356i	0.967811	−	0.251678i	$-0.0809823\pi$
$368$		0			0
$369$	−174.456			−0.472781
$370$		0			0
$371$		−	27.8152i		−	0.0749735i
$372$		0			0
$373$	−58.1492			−0.155896			−0.0779480	−	0.996957i	$-0.524837\pi$
$373$	−58.1492			−0.155896			−0.0779480	+	0.996957i	$0.524837\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	243.924			0.647013
$378$		0			0
$379$	−25.5878			−0.0675139			−0.0337569	−	0.999430i	$-0.510747\pi$
$379$	−25.5878			−0.0675139			−0.0337569	+	0.999430i	$0.510747\pi$
$380$		0			0
$381$	−446.408			−1.17167
$382$		0			0
$383$			441.976i			1.15398i	0.816750	+	0.576992i	$0.195773\pi$
$383$			441.976i			1.15398i	−0.816750	+	0.576992i	$0.804227\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			54.1160i			0.139835i
$388$		0			0
$389$		−	11.5630i		−	0.0297250i	−0.999890	−	0.0148625i	$-0.995269\pi$
$389$		−	11.5630i		−	0.0297250i	0.999890	−	0.0148625i	$-0.00473106\pi$
$390$		0			0
$391$	−52.9328			−0.135378
$392$		0			0
$393$		−	220.742i		−	0.561685i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	110.621			0.278641			0.139321	−	0.990247i	$-0.455508\pi$
$397$	110.621			0.278641			0.139321	+	0.990247i	$0.455508\pi$
$398$		0			0
$399$		−	51.4176i		−	0.128866i
$400$		0			0
$401$		−	556.915i		−	1.38881i	−0.719582	−	0.694407i	$-0.755667\pi$
$401$		−	556.915i		−	1.38881i	0.719582	−	0.694407i	$-0.244333\pi$
$402$		0			0
$403$	−54.7668	−	185.426i	−0.135898	−	0.460114i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	461.105			1.13294
$408$		0			0
$409$		−	207.687i		−	0.507793i	−0.967231	−	0.253896i	$-0.918288\pi$
$409$		−	207.687i		−	0.507793i	0.967231	−	0.253896i	$-0.0817122\pi$
$410$		0			0
$411$	−230.514			−0.560861
$412$		0			0
$413$	−90.8386			−0.219948
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−45.9837			−0.110273
$418$		0			0
$419$	−768.752			−1.83473			−0.917365	−	0.398048i	$-0.869688\pi$
$419$	−768.752			−1.83473			−0.917365	+	0.398048i	$0.869688\pi$
$420$		0			0
$421$	275.974			0.655520			0.327760	−	0.944761i	$-0.393706\pi$
$421$	275.974			0.655520			0.327760	+	0.944761i	$0.393706\pi$
$422$		0			0
$423$	169.705			0.401194
$424$		0			0
$425$		0			0
$426$		0			0
$427$			68.9993i			0.161591i
$428$		0			0
$429$			134.200i			0.312820i
$430$		0			0
$431$	−45.2542			−0.104998			−0.0524990	−	0.998621i	$-0.516719\pi$
$431$	−45.2542			−0.104998			−0.0524990	+	0.998621i	$0.516719\pi$
$432$		0			0
$433$			220.747i			0.509807i	0.966966	+	0.254904i	$0.0820438\pi$
$433$			220.747i			0.509807i	−0.966966	+	0.254904i	$0.917956\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			474.176i			1.08507i
$438$		0			0
$439$	425.970			0.970318			0.485159	−	0.874426i	$-0.338762\pi$
$439$	425.970			0.970318			0.485159	+	0.874426i	$0.338762\pi$
$440$		0			0
$441$	−226.783			−0.514248
$442$		0			0
$443$	79.2061			0.178795			0.0893974	−	0.995996i	$-0.471506\pi$
$443$	79.2061			0.178795			0.0893974	+	0.995996i	$0.471506\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			82.0533i			0.183564i
$448$		0			0
$449$			706.106i			1.57262i	0.617833	+	0.786309i	$0.288011\pi$
$449$			706.106i			1.57262i	−0.617833	+	0.786309i	$0.711989\pi$
$450$		0			0
$451$			380.821i			0.844393i
$452$		0			0
$453$	−299.807			−0.661825
$454$		0			0
$455$		0			0
$456$		0			0
$457$			158.503i			0.346833i	0.984849	+	0.173416i	$0.0554807\pi$
$457$			158.503i			0.346833i	−0.984849	+	0.173416i	$0.944519\pi$
$458$		0			0
$459$	56.4164			0.122912
$460$		0			0
$461$		−	796.474i		−	1.72771i	−0.503741	−	0.863855i	$-0.668043\pi$
$461$		−	796.474i		−	1.72771i	0.503741	−	0.863855i	$-0.331957\pi$
$462$		0			0
$463$		−	809.006i		−	1.74731i	−0.486544	−	0.873656i	$-0.661743\pi$
$463$		−	809.006i		−	1.74731i	0.486544	−	0.873656i	$-0.338257\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	244.554			0.523669			0.261835	−	0.965113i	$-0.415672\pi$
$467$	244.554			0.523669			0.261835	+	0.965113i	$0.415672\pi$
$468$		0			0
$469$	−62.9128			−0.134143
$470$		0			0
$471$		−	461.529i		−	0.979892i
$472$		0			0
$473$	118.130			0.249747
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	95.3891i		−	0.199977i
$478$		0			0
$479$	−32.2869			−0.0674048			−0.0337024	−	0.999432i	$-0.510730\pi$
$479$	−32.2869			−0.0674048			−0.0337024	+	0.999432i	$0.510730\pi$
$480$		0			0
$481$	−273.167			−0.567914
$482$		0			0
$483$	76.1876			0.157738
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	380.424i		−	0.781158i	−0.920569	−	0.390579i	$-0.872275\pi$
$487$		−	380.424i		−	0.781158i	0.920569	−	0.390579i	$-0.127725\pi$
$488$		0			0
$489$		−	49.5497i		−	0.101329i
$490$		0			0
$491$			288.334i			0.587239i	0.955922	+	0.293619i	$0.0948598\pi$
$491$			288.334i			0.587239i	−0.955922	+	0.293619i	$0.905140\pi$
$492$		0			0
$493$	−78.1002			−0.158418
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−133.203			−0.268015
$498$		0			0
$499$			55.5888i			0.111400i	0.998448	+	0.0557002i	$0.0177391\pi$
$499$			55.5888i			0.111400i	−0.998448	+	0.0557002i	$0.982261\pi$
$500$		0			0
$501$	−277.041			−0.552975
$502$		0			0
$503$	326.896			0.649893			0.324946	−	0.945732i	$-0.394654\pi$
$503$	326.896			0.649893			0.324946	+	0.945732i	$0.394654\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			265.899i			0.524456i
$508$		0			0
$509$		−	242.908i		−	0.477227i	−0.971115	−	0.238613i	$-0.923307\pi$
$509$		−	242.908i		−	0.477227i	0.971115	−	0.238613i	$-0.0766928\pi$
$510$		0			0
$511$			135.821i			0.265795i
$512$		0			0
$513$		−	505.382i		−	0.985151i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	370.450i		−	0.716539i
$518$		0			0
$519$		−	537.742i		−	1.03611i
$520$		0			0
$521$	371.942			0.713900			0.356950	−	0.934123i	$-0.383817\pi$
$521$	371.942			0.713900			0.356950	+	0.934123i	$0.383817\pi$
$522$		0			0
$523$			227.375i			0.434751i	0.976088	+	0.217376i	$0.0697496\pi$
$523$			227.375i			0.434751i	−0.976088	+	0.217376i	$0.930250\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	17.5354	+	59.3702i	0.0332740	+	0.112657i
$528$		0			0
$529$	−173.605			−0.328177
$530$		0			0
$531$	−311.521			−0.586669
$532$		0			0
$533$		−	225.605i		−	0.423274i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	553.663			1.03103
$538$		0			0
$539$			495.047i			0.918454i
$540$		0			0
$541$	−136.811			−0.252885			−0.126442	−	0.991974i	$-0.540356\pi$
$541$	−136.811			−0.252885			−0.126442	+	0.991974i	$0.540356\pi$
$542$		0			0
$543$	526.774			0.970117
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−556.341			−1.01708			−0.508538	−	0.861039i	$-0.669814\pi$
$547$	−556.341			−1.01708			−0.508538	+	0.861039i	$0.669814\pi$
$548$		0			0
$549$			236.626i			0.431012i
$550$		0			0
$551$			699.628i			1.26974i
$552$		0			0
$553$			112.402i			0.203258i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			330.379i			0.593140i	0.955011	+	0.296570i	$0.0958428\pi$
$557$			330.379i			0.593140i	−0.955011	+	0.296570i	$0.904157\pi$
$558$		0			0
$559$	−69.9824			−0.125192
$560$		0			0
$561$		−	42.9684i		−	0.0765926i
$562$		0			0
$563$	−229.619			−0.407850			−0.203925	−	0.978987i	$-0.565370\pi$
$563$	−229.619			−0.407850			−0.203925	+	0.978987i	$0.565370\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−20.1579			−0.0355518
$568$		0			0
$569$		−	309.351i		−	0.543675i	−0.962343	−	0.271838i	$-0.912369\pi$
$569$		−	309.351i		−	0.543675i	0.962343	−	0.271838i	$-0.0876314\pi$
$570$		0			0
$571$		−	276.169i		−	0.483659i	−0.970319	−	0.241830i	$-0.922252\pi$
$571$		−	276.169i		−	0.483659i	0.970319	−	0.241830i	$-0.0777475\pi$
$572$		0			0
$573$			114.574i			0.199955i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	873.227			1.51339			0.756696	−	0.653767i	$-0.226812\pi$
$577$	873.227			1.51339			0.756696	+	0.653767i	$0.226812\pi$
$578$		0			0
$579$		−	501.298i		−	0.865799i
$580$		0			0
$581$			27.2861i			0.0469640i
$582$		0			0
$583$	−208.225			−0.357162
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	116.004i		−	0.197622i	−0.995106	−	0.0988112i	$-0.968496\pi$
$587$		−	116.004i		−	0.197622i	0.995106	−	0.0988112i	$-0.0315040\pi$
$588$		0			0
$589$	531.843	−	157.083i	0.902959	−	0.266695i
$590$		0			0
$591$	144.555			0.244594
$592$		0			0
$593$	560.397			0.945020			0.472510	−	0.881325i	$-0.343348\pi$
$593$	560.397			0.945020			0.472510	+	0.881325i	$0.343348\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−73.9329			−0.123841
$598$		0			0
$599$	307.595			0.513515			0.256757	−	0.966476i	$-0.417346\pi$
$599$	307.595			0.513515			0.256757	+	0.966476i	$0.417346\pi$
$600$		0			0
$601$			908.857i			1.51224i	0.654432	+	0.756121i	$0.272908\pi$
$601$			908.857i			1.51224i	−0.654432	+	0.756121i	$0.727092\pi$
$602$		0			0
$603$	−215.753			−0.357799
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−768.648			−1.26631			−0.633153	−	0.774026i	$-0.718240\pi$
$607$	−768.648			−1.26631			−0.633153	+	0.774026i	$0.718240\pi$
$608$		0			0
$609$	112.412			0.184584
$610$		0			0
$611$			219.461i			0.359184i
$612$		0			0
$613$			310.570i			0.506639i	0.967383	+	0.253320i	$0.0815224\pi$
$613$			310.570i			0.506639i	−0.967383	+	0.253320i	$0.918478\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−358.529			−0.581084			−0.290542	−	0.956862i	$-0.593836\pi$
$617$	−358.529			−0.581084			−0.290542	+	0.956862i	$0.593836\pi$
$618$		0			0
$619$		−	137.717i		−	0.222484i	−0.993793	−	0.111242i	$-0.964517\pi$
$619$		−	137.717i		−	0.222484i	0.993793	−	0.111242i	$-0.0354828\pi$
$620$		0			0
$621$	748.846			1.20587
$622$		0			0
$623$		−	76.2435i		−	0.122381i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−384.914			−0.613898
$628$		0			0
$629$	87.4632			0.139051
$630$		0			0
$631$			496.857i			0.787412i	0.919236	+	0.393706i	$0.128807\pi$
$631$			496.857i			0.787412i	−0.919236	+	0.393706i	$0.871193\pi$
$632$		0			0
$633$		−	259.028i		−	0.409207i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	293.274i		−	0.460399i
$638$		0			0
$639$	−456.807			−0.714878
$640$		0			0
$641$		−	745.808i		−	1.16351i	−0.813365	−	0.581753i	$-0.802367\pi$
$641$		−	745.808i		−	1.16351i	0.813365	−	0.581753i	$-0.197633\pi$
$642$		0			0
$643$			804.010i			1.25040i	0.780463	+	0.625202i	$0.214983\pi$
$643$			804.010i			1.25040i	−0.780463	+	0.625202i	$0.785017\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			376.048i			0.581218i	0.956842	+	0.290609i	$0.0938580\pi$
$647$			376.048i			0.581218i	−0.956842	+	0.290609i	$0.906142\pi$
$648$		0			0
$649$			680.021i			1.04780i
$650$		0			0
$651$	−25.2392	−	85.4532i	−0.0387698	−	0.131265i
$652$		0			0
$653$	−14.2731			−0.0218578			−0.0109289	−	0.999940i	$-0.503479\pi$
$653$	−14.2731			−0.0218578			−0.0109289	+	0.999940i	$0.503479\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			465.785i			0.708957i
$658$		0			0
$659$	−420.989			−0.638830			−0.319415	−	0.947615i	$-0.603486\pi$
$659$	−420.989			−0.638830			−0.319415	+	0.947615i	$0.603486\pi$
$660$		0			0
$661$	753.341			1.13970			0.569850	−	0.821749i	$-0.307001\pi$
$661$	753.341			1.13970			0.569850	+	0.821749i	$0.307001\pi$
$662$		0			0
$663$			25.4552i			0.0383940i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1036.67			−1.55422
$668$		0			0
$669$	730.345			1.09170
$670$		0			0
$671$	516.532			0.769794
$672$		0			0
$673$			916.367i			1.36162i	0.732462	+	0.680808i	$0.238371\pi$
$673$			916.367i			1.36162i	−0.732462	+	0.680808i	$0.761629\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	433.168i		−	0.639835i	−0.947445	−	0.319917i	$-0.896345\pi$
$677$		−	433.168i		−	0.639835i	0.947445	−	0.319917i	$-0.103655\pi$
$678$		0			0
$679$	131.583			0.193790
$680$		0			0
$681$			111.250i			0.163362i
$682$		0			0
$683$	−287.577			−0.421050			−0.210525	−	0.977589i	$-0.567517\pi$
$683$	−287.577			−0.421050			−0.210525	+	0.977589i	$0.567517\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−611.951			−0.890759
$688$		0			0
$689$	123.356			0.179037
$690$		0			0
$691$	26.1197			0.0377998			0.0188999	−	0.999821i	$-0.493984\pi$
$691$	26.1197			0.0377998			0.0188999	+	0.999821i	$0.493984\pi$
$692$		0			0
$693$		−	71.4072i		−	0.103041i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			72.2348i			0.103637i
$698$		0			0
$699$		−	249.317i		−	0.356677i
$700$		0			0
$701$	−283.101			−0.403852			−0.201926	−	0.979401i	$-0.564720\pi$
$701$	−283.101			−0.403852			−0.201926	+	0.979401i	$0.564720\pi$
$702$		0			0
$703$		−	783.502i		−	1.11451i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	11.3103			0.0159976
$708$		0			0
$709$		−	214.425i		−	0.302434i	−0.988501	−	0.151217i	$-0.951681\pi$
$709$		−	214.425i		−	0.302434i	0.988501	−	0.151217i	$-0.0483192\pi$
$710$		0			0
$711$			385.469i			0.542151i
$712$		0			0
$713$	232.757	+	788.053i	0.326447	+	1.10526i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	757.001			1.05579
$718$		0			0
$719$			957.448i			1.33164i	0.746113	+	0.665819i	$0.231918\pi$
$719$			957.448i			1.33164i	−0.746113	+	0.665819i	$0.768082\pi$
$720$		0			0
$721$	−198.154			−0.274833
$722$		0			0
$723$	−326.384			−0.451430
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−1314.74			−1.80844			−0.904220	−	0.427066i	$-0.859547\pi$
$727$	−1314.74			−1.80844			−0.904220	+	0.427066i	$0.859547\pi$
$728$		0			0
$729$	−588.786			−0.807663
$730$		0			0
$731$	22.4071			0.0306527
$732$		0			0
$733$	−1277.01			−1.74217			−0.871086	−	0.491130i	$-0.836584\pi$
$733$	−1277.01			−1.74217			−0.871086	+	0.491130i	$0.836584\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			470.968i			0.639034i
$738$		0			0
$739$			203.193i			0.274956i	0.990505	+	0.137478i	$0.0438997\pi$
$739$			203.193i			0.274956i	−0.990505	+	0.137478i	$0.956100\pi$
$740$		0			0
$741$	228.030			0.307733
$742$		0			0
$743$		−	470.952i		−	0.633851i	−0.948450	−	0.316926i	$-0.897349\pi$
$743$		−	470.952i		−	0.633851i	0.948450	−	0.316926i	$-0.102651\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			93.5746i			0.125267i
$748$		0			0
$749$	111.705			0.149139
$750$		0			0
$751$	656.317			0.873924			0.436962	−	0.899480i	$-0.356054\pi$
$751$	656.317			0.873924			0.436962	+	0.899480i	$0.356054\pi$
$752$		0			0
$753$	354.472			0.470746
$754$		0			0
$755$		0			0
$756$		0			0
$757$			266.647i			0.352242i	0.984369	+	0.176121i	$0.0563550\pi$
$757$			266.647i			0.352242i	−0.984369	+	0.176121i	$0.943645\pi$
$758$		0			0
$759$		−	570.343i		−	0.751440i
$760$		0			0
$761$			834.798i			1.09697i	0.836159	+	0.548487i	$0.184796\pi$
$761$			834.798i			1.09697i	−0.836159	+	0.548487i	$0.815204\pi$
$762$		0			0
$763$	111.463			0.146086
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	402.857i		−	0.525237i
$768$		0			0
$769$	599.143			0.779120			0.389560	−	0.921001i	$-0.372627\pi$
$769$	599.143			0.779120			0.389560	+	0.921001i	$0.372627\pi$
$770$		0			0
$771$			413.419i			0.536212i
$772$		0			0
$773$			121.997i			0.157823i	0.996882	+	0.0789114i	$0.0251444\pi$
$773$			121.997i			0.157823i	−0.996882	+	0.0789114i	$0.974856\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−125.888			−0.162018
$778$		0			0
$779$	647.085			0.830661
$780$		0			0
$781$			997.166i			1.27678i
$782$		0			0
$783$	1104.89			1.41110
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	1237.53i		−	1.57247i	−0.617928	−	0.786235i	$-0.712028\pi$
$787$		−	1237.53i		−	1.57247i	0.617928	−	0.786235i	$-0.287972\pi$
$788$		0			0
$789$	382.778			0.485143
$790$		0			0
$791$	−187.917			−0.237569
$792$		0			0
$793$	−306.003			−0.385880
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1171.84i			1.47031i	0.677898	+	0.735156i	$0.262891\pi$
$797$			1171.84i			1.47031i	−0.677898	+	0.735156i	$0.737109\pi$
$798$		0			0
$799$		−	70.2677i		−	0.0879446i
$800$		0			0
$801$		−	261.469i		−	0.326428i
$802$		0			0
$803$	1016.76			1.26621
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−953.677			−1.18176
$808$		0			0
$809$		−	1535.83i		−	1.89844i	−0.314619	−	0.949218i	$-0.601877\pi$
$809$		−	1535.83i		−	1.89844i	0.314619	−	0.949218i	$-0.398123\pi$
$810$		0			0
$811$	1477.37			1.82166			0.910832	−	0.412778i	$-0.135441\pi$
$811$	1477.37			1.82166			0.910832	+	0.412778i	$0.135441\pi$
$812$		0			0
$813$	628.629			0.773222
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	200.725i		−	0.245685i
$818$		0			0
$819$			42.3029i			0.0516519i
$820$		0			0
$821$			1010.64i			1.23099i	0.788140	+	0.615496i	$0.211044\pi$
$821$			1010.64i			1.23099i	−0.788140	+	0.615496i	$0.788956\pi$
$822$		0			0
$823$			444.224i			0.539762i	0.962894	+	0.269881i	$0.0869844\pi$
$823$			444.224i			0.539762i	−0.962894	+	0.269881i	$0.913016\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			355.923i			0.430378i	0.976572	+	0.215189i	$0.0690368\pi$
$827$			355.923i			0.430378i	−0.976572	+	0.215189i	$0.930963\pi$
$828$		0			0
$829$			109.963i			0.132645i	0.997798	+	0.0663224i	$0.0211266\pi$
$829$			109.963i			0.132645i	−0.997798	+	0.0663224i	$0.978873\pi$
$830$		0			0
$831$	472.650			0.568772
$832$		0			0
$833$			93.9013i			0.112727i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−248.075	−	839.917i	−0.296386	−	1.00349i
$838$		0			0
$839$	−557.732			−0.664758			−0.332379	−	0.943146i	$-0.607851\pi$
$839$	−557.732			−0.664758			−0.332379	+	0.943146i	$0.607851\pi$
$840$		0			0
$841$	−688.561			−0.818741
$842$		0			0
$843$		−	600.520i		−	0.712360i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	14.2922			0.0168739
$848$		0			0
$849$		−	491.995i		−	0.579500i
$850$		0			0
$851$	1160.95			1.36421
$852$		0			0
$853$	1076.11			1.26156			0.630778	−	0.775964i	$-0.282736\pi$
$853$	1076.11			1.26156			0.630778	+	0.775964i	$0.282736\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	915.926			1.06876			0.534379	−	0.845245i	$-0.320545\pi$
$857$	915.926			1.06876			0.534379	+	0.845245i	$0.320545\pi$
$858$		0			0
$859$			624.874i			0.727443i	0.931508	+	0.363722i	$0.118494\pi$
$859$			624.874i			0.727443i	−0.931508	+	0.363722i	$0.881506\pi$
$860$		0			0
$861$		−	103.970i		−	0.120754i
$862$		0			0
$863$			1266.28i			1.46730i	0.679526	+	0.733652i	$0.262186\pi$
$863$			1266.28i			1.46730i	−0.679526	+	0.733652i	$0.737814\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			582.507i			0.671865i
$868$		0			0
$869$	841.443			0.968289
$870$		0			0
$871$		−	279.010i		−	0.320333i
$872$		0			0
$873$	451.251			0.516897
$874$		0			0
$875$		0			0
$876$		0			0
$877$	487.870			0.556295			0.278147	−	0.960538i	$-0.410280\pi$
$877$	487.870			0.556295			0.278147	+	0.960538i	$0.410280\pi$
$878$		0			0
$879$		−	659.059i		−	0.749783i
$880$		0			0
$881$			270.653i			0.307211i	0.988132	+	0.153606i	$0.0490885\pi$
$881$			270.653i			0.307211i	−0.988132	+	0.153606i	$0.950911\pi$
$882$		0			0
$883$		−	192.419i		−	0.217915i	−0.994046	−	0.108958i	$-0.965249\pi$
$883$		−	192.419i		−	0.217915i	0.994046	−	0.108958i	$-0.0347513\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−241.910			−0.272728			−0.136364	−	0.990659i	$-0.543542\pi$
$887$	−241.910			−0.272728			−0.136364	+	0.990659i	$0.543542\pi$
$888$		0			0
$889$			307.174i			0.345528i
$890$		0			0
$891$			150.903i			0.169363i
$892$		0			0
$893$	−629.463			−0.704886
$894$		0			0
$895$		0			0
$896$		0			0
$897$			337.881i			0.376679i
$898$		0			0
$899$	343.423	+	1162.74i	0.382006	+	1.29337i
$900$		0			0
$901$	−39.4966			−0.0438364
$902$		0			0
$903$	−32.2512			−0.0357156
$904$		0			0
$905$		0			0
$906$		0			0
$907$	154.601			0.170454			0.0852268	−	0.996362i	$-0.472839\pi$
$907$	154.601			0.170454			0.0852268	+	0.996362i	$0.472839\pi$
$908$		0			0
$909$	38.7876			0.0426706
$910$		0			0
$911$		−	1088.77i		−	1.19514i	−0.801817	−	0.597569i	$-0.796133\pi$
$911$		−	1088.77i		−	1.19514i	0.801817	−	0.597569i	$-0.203867\pi$
$912$		0			0
$913$	204.265			0.223729
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−151.893			−0.165642
$918$		0			0
$919$	478.821			0.521024			0.260512	−	0.965471i	$-0.416109\pi$
$919$	478.821			0.521024			0.260512	+	0.965471i	$0.416109\pi$
$920$		0			0
$921$		−	315.434i		−	0.342491i
$922$		0			0
$923$		−	590.739i		−	0.640020i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−679.549			−0.733063
$928$		0			0
$929$			1586.32i			1.70756i	0.520634	+	0.853780i	$0.325696\pi$
$929$			1586.32i			1.70756i	−0.520634	+	0.853780i	$0.674304\pi$
$930$		0			0
$931$	841.175			0.903518
$932$		0			0
$933$		−	865.557i		−	0.927714i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1495.72			1.59629			0.798143	−	0.602468i	$-0.205816\pi$
$937$	1495.72			1.59629			0.798143	+	0.602468i	$0.205816\pi$
$938$		0			0
$939$	294.707			0.313852
$940$		0			0
$941$			945.614i			1.00490i	0.864605	+	0.502452i	$0.167569\pi$
$941$			945.614i			1.00490i	−0.864605	+	0.502452i	$0.832431\pi$
$942$		0			0
$943$			958.812i			1.01677i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			277.712i			0.293255i	0.989192	+	0.146627i	$0.0468418\pi$
$947$			277.712i			0.293255i	−0.989192	+	0.146627i	$0.953158\pi$
$948$		0			0
$949$	−602.349			−0.634720
$950$		0			0
$951$			887.667i			0.933403i
$952$		0			0
$953$		−	1008.40i		−	1.05813i	−0.848580	−	0.529067i	$-0.822542\pi$
$953$		−	1008.40i		−	1.05813i	0.848580	−	0.529067i	$-0.177458\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	841.519i		−	0.879330i
$958$		0			0
$959$			158.617i			0.165399i
$960$		0			0
$961$	806.786	−	522.127i	0.839528	−	0.543317i
$962$		0			0
$963$	383.080			0.397799
$964$		0			0
$965$		0			0
$966$		0			0
$967$			401.979i			0.415697i	0.978161	+	0.207848i	$0.0666460\pi$
$967$			401.979i			0.415697i	−0.978161	+	0.207848i	$0.933354\pi$
$968$		0			0
$969$	−73.0112			−0.0753470
$970$		0			0
$971$	−560.114			−0.576842			−0.288421	−	0.957504i	$-0.593130\pi$
$971$	−560.114			−0.576842			−0.288421	+	0.957504i	$0.593130\pi$
$972$		0			0
$973$			31.6415i			0.0325196i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1362.67			1.39475			0.697377	−	0.716705i	$-0.254350\pi$
$977$	1362.67			1.39475			0.697377	+	0.716705i	$0.254350\pi$
$978$		0			0
$979$	−570.762			−0.583005
$980$		0			0
$981$	382.252			0.389655
$982$		0			0
$983$			810.338i			0.824352i	0.911104	+	0.412176i	$0.135231\pi$
$983$			810.338i			0.824352i	−0.911104	+	0.412176i	$0.864769\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			101.138i			0.102470i
$988$		0			0
$989$	297.422			0.300730
$990$		0			0
$991$		−	572.460i		−	0.577659i	−0.957381	−	0.288829i	$-0.906734\pi$
$991$		−	572.460i		−	0.577659i	0.957381	−	0.288829i	$-0.0932661\pi$
$992$		0			0
$993$	1288.09			1.29717
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1308.76			−1.31270			−0.656351	−	0.754456i	$-0.727901\pi$
$997$	−1308.76			−1.31270			−0.656351	+	0.754456i	$0.727901\pi$
$998$		0			0
$999$	−1237.35			−1.23859

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3100.3.d.e.1301.14		20
5.2	odd	4		3100.3.f.c.1549.27		40
5.3	odd	4		3100.3.f.c.1549.14		40
5.4	even	2		620.3.d.a.61.7	✓	20
31.30	odd	2	inner	3100.3.d.e.1301.7		20
155.92	even	4		3100.3.f.c.1549.13		40
155.123	even	4		3100.3.f.c.1549.28		40
155.154	odd	2		620.3.d.a.61.14	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
620.3.d.a.61.7	✓	20	5.4	even	2
620.3.d.a.61.14	yes	20	155.154	odd	2
3100.3.d.e.1301.7		20	31.30	odd	2	inner
3100.3.d.e.1301.14		20	1.1	even	1	trivial
3100.3.f.c.1549.13		40	155.92	even	4
3100.3.f.c.1549.14		40	5.3	odd	4
3100.3.f.c.1549.27		40	5.2	odd	4
3100.3.f.c.1549.28		40	155.123	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:	\( N \)	\(=\)	\( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3100.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.04380i q^{3} +1.40634 q^{7} +4.82290 q^{9} +O(q^{10})\)	\(q+2.04380i q^{3} +1.40634 q^{7} +4.82290 q^{9} -10.5279i q^{11} +6.23693i q^{13} -1.99696i q^{17} -17.8889 q^{19} +2.87428i q^{21} -26.5067i q^{23} +28.2512i q^{27} -39.1096i q^{29} +(-29.7303 + 8.78105i) q^{31} +21.5169 q^{33} +43.7982i q^{37} -12.7470 q^{39} -36.1724 q^{41} +11.2206i q^{43} +35.1874 q^{47} -47.0222 q^{49} +4.08137 q^{51} -19.7784i q^{53} -36.5612i q^{57} -64.5921 q^{59} +49.0630i q^{61} +6.78265 q^{63} -44.7351 q^{67} +54.1743 q^{69} -94.7163 q^{71} +96.5778i q^{73} -14.8059i q^{77} +79.9248i q^{79} -14.3336 q^{81} +19.4022i q^{83} +79.9320 q^{87} -54.2141i q^{89} +8.77126i q^{91} +(-17.9467 - 60.7627i) q^{93} +93.5642 q^{97} -50.7751i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{7} - 48 q^{9}+O(q^{10}) \) 20 * q - 12 * q^7 - 48 * q^9	\( 20 q - 12 q^{7} - 48 q^{9} + 60 q^{19} + 8 q^{31} - 68 q^{33} + 28 q^{39} - 80 q^{41} + 48 q^{47} + 84 q^{49} + 344 q^{51} + 160 q^{59} + 232 q^{63} + 180 q^{67} - 140 q^{69} - 108 q^{71} + 336 q^{81} + 236 q^{87} + 332 q^{93} + 112 q^{97}+O(q^{100}) \) 20 * q - 12 * q^7 - 48 * q^9 + 60 * q^19 + 8 * q^31 - 68 * q^33 + 28 * q^39 - 80 * q^41 + 48 * q^47 + 84 * q^49 + 344 * q^51 + 160 * q^59 + 232 * q^63 + 180 * q^67 - 140 * q^69 - 108 * q^71 + 336 * q^81 + 236 * q^87 + 332 * q^93 + 112 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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