LMFDB - Embedded newform 3100.3.f.b.1549.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3100,3,Mod(1549,3100)]

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, 1, 1]))

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

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

chi := DirichletCharacter("3100.1549");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3100 = 2^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3100.f (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:	$84.4688819517$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.256992219136.8
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 105x^{4} + 64 $ x^8 + 105*x^4 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 124)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1549.5
Root			$0.625703 + 0.625703i$ of defining polynomial
Character	$\chi$	$=$	3100.1549
Dual form			3100.3.f.b.1549.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.25141 q^{3} -6.21699i q^{7} -7.43398 q^{9} +O(q^{10})$	$q+1.25141 q^{3} -6.21699i q^{7} -7.43398 q^{9} -14.0370i q^{11} +19.3142 q^{13} +28.3456 q^{17} -12.6510 q^{19} -7.77998i q^{21} +5.27717 q^{23} -20.5656 q^{27} -34.3311i q^{29} +(23.4340 + 20.2941i) q^{31} -17.5660i q^{33} +34.6026 q^{37} +24.1699 q^{39} -59.5189 q^{41} +1.25141 q^{43} +22.1699i q^{47} +10.3490 q^{49} +35.4718 q^{51} -31.5567 q^{53} -15.8315 q^{57} +41.9529 q^{59} -46.8451i q^{61} +46.2170i q^{63} -45.1320i q^{67} +6.60389 q^{69} +67.2549 q^{71} -48.6396 q^{73} -87.2680 q^{77} +151.196i q^{79} +41.1699 q^{81} -87.7048 q^{83} -42.9621i q^{87} -98.0939i q^{89} -120.076i q^{91} +(29.3254 + 25.3961i) q^{93} -100.821i q^{97} +104.351i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{9}+O(q^{10}) $ 8 * q + 16 * q^9	$ 8 q + 16 q^{9} + 12 q^{19} + 112 q^{31} - 184 q^{39} - 212 q^{41} + 196 q^{49} - 320 q^{51} - 4 q^{59} - 400 q^{69} - 28 q^{71} - 48 q^{81}+O(q^{100}) $ 8 * q + 16 * q^9 + 12 * q^19 + 112 * q^31 - 184 * q^39 - 212 * q^41 + 196 * q^49 - 320 * q^51 - 4 * q^59 - 400 * q^69 - 28 * q^71 - 48 * q^81

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
$3$	1.25141			0.417136			0.208568	−	0.978008i	$-0.433120\pi$
$3$	1.25141			0.417136			0.208568	+	0.978008i	$0.433120\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	6.21699i		−	0.888142i	−0.895992	−	0.444071i	$-0.853534\pi$
$7$		−	6.21699i		−	0.888142i	0.895992	−	0.444071i	$-0.146466\pi$
$8$		0			0
$9$	−7.43398			−0.825998
$10$		0			0
$11$		−	14.0370i		−	1.27609i	−0.769998	−	0.638046i	$-0.779743\pi$
$11$		−	14.0370i		−	1.27609i	0.769998	−	0.638046i	$-0.220257\pi$
$12$		0			0
$13$	19.3142			1.48571			0.742853	−	0.669454i	$-0.233472\pi$
$13$	19.3142			1.48571			0.742853	+	0.669454i	$0.233472\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	28.3456			1.66739			0.833694	−	0.552227i	$-0.186222\pi$
$17$	28.3456			1.66739			0.833694	+	0.552227i	$0.186222\pi$
$18$		0			0
$19$	−12.6510			−0.665841			−0.332920	−	0.942955i	$-0.608034\pi$
$19$	−12.6510			−0.665841			−0.332920	+	0.942955i	$0.608034\pi$
$20$		0			0
$21$		−	7.77998i		−	0.370475i
$22$		0			0
$23$	5.27717			0.229442			0.114721	−	0.993398i	$-0.463403\pi$
$23$	5.27717			0.229442			0.114721	+	0.993398i	$0.463403\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−20.5656			−0.761689
$28$		0			0
$29$		−	34.3311i		−	1.18383i	−0.806000	−	0.591915i	$-0.798372\pi$
$29$		−	34.3311i		−	1.18383i	0.806000	−	0.591915i	$-0.201628\pi$
$30$		0			0
$31$	23.4340	+	20.2941i	0.755935	+	0.654647i
$31$	23.4340	+	20.2941i	0.755935	+	0.654647i
$32$		0			0
$33$		−	17.5660i		−	0.532304i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	34.6026			0.935206			0.467603	−	0.883939i	$-0.345118\pi$
$37$	34.6026			0.935206			0.467603	+	0.883939i	$0.345118\pi$
$38$		0			0
$39$	24.1699			0.619741
$40$		0			0
$41$	−59.5189			−1.45168			−0.725841	−	0.687863i	$-0.758549\pi$
$41$	−59.5189			−1.45168			−0.725841	+	0.687863i	$0.758549\pi$
$42$		0			0
$43$	1.25141			0.0291025			0.0145512	−	0.999894i	$-0.495368\pi$
$43$	1.25141			0.0291025			0.0145512	+	0.999894i	$0.495368\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			22.1699i			0.471700i	0.971789	+	0.235850i	$0.0757874\pi$
$47$			22.1699i			0.471700i	−0.971789	+	0.235850i	$0.924213\pi$
$48$		0			0
$49$	10.3490			0.211205
$50$		0			0
$51$	35.4718			0.695526
$52$		0			0
$53$	−31.5567			−0.595410			−0.297705	−	0.954658i	$-0.596221\pi$
$53$	−31.5567			−0.595410			−0.297705	+	0.954658i	$0.596221\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−15.8315			−0.277746
$58$		0			0
$59$	41.9529			0.711066			0.355533	−	0.934664i	$-0.384299\pi$
$59$	41.9529			0.711066			0.355533	+	0.934664i	$0.384299\pi$
$60$		0			0
$61$		−	46.8451i		−	0.767953i	−0.923343	−	0.383977i	$-0.874554\pi$
$61$		−	46.8451i		−	0.767953i	0.923343	−	0.383977i	$-0.125446\pi$
$62$		0			0
$63$			46.2170i			0.733603i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	45.1320i		−	0.673613i	−0.941574	−	0.336806i	$-0.890653\pi$
$67$		−	45.1320i		−	0.673613i	0.941574	−	0.336806i	$-0.109347\pi$
$68$		0			0
$69$	6.60389			0.0957085
$70$		0			0
$71$	67.2549			0.947252			0.473626	−	0.880726i	$-0.342945\pi$
$71$	67.2549			0.947252			0.473626	+	0.880726i	$0.342945\pi$
$72$		0			0
$73$	−48.6396			−0.666296			−0.333148	−	0.942874i	$-0.608111\pi$
$73$	−48.6396			−0.666296			−0.333148	+	0.942874i	$0.608111\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−87.2680			−1.13335
$78$		0			0
$79$			151.196i			1.91387i	0.290295	+	0.956937i	$0.406246\pi$
$79$			151.196i			1.91387i	−0.290295	+	0.956937i	$0.593754\pi$
$80$		0			0
$81$	41.1699			0.508270
$82$		0			0
$83$	−87.7048			−1.05668			−0.528342	−	0.849032i	$-0.677186\pi$
$83$	−87.7048			−1.05668			−0.528342	+	0.849032i	$0.677186\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	42.9621i		−	0.493818i
$88$		0			0
$89$		−	98.0939i		−	1.10218i	−0.834446	−	0.551089i	$-0.814212\pi$
$89$		−	98.0939i		−	1.10218i	0.834446	−	0.551089i	$-0.185788\pi$
$90$		0			0
$91$		−	120.076i		−	1.31952i
$92$		0			0
$93$	29.3254	+	25.3961i	0.315327	+	0.273076i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	100.821i		−	1.03939i	−0.854352	−	0.519695i	$-0.826045\pi$
$97$		−	100.821i		−	1.03939i	0.854352	−	0.519695i	$-0.173955\pi$
$98$		0			0
$99$			104.351i			1.05405i
$100$		0			0
$101$	−17.6131			−0.174387			−0.0871936	−	0.996191i	$-0.527790\pi$
$101$	−17.6131			−0.174387			−0.0871936	+	0.996191i	$0.527790\pi$
$102$		0			0
$103$		−	41.6131i		−	0.404011i	−0.979384	−	0.202005i	$-0.935254\pi$
$103$		−	41.6131i		−	0.404011i	0.979384	−	0.202005i	$-0.0647458\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			91.3490i			0.853729i	0.904316	+	0.426865i	$0.140382\pi$
$107$			91.3490i			0.853729i	−0.904316	+	0.426865i	$0.859618\pi$
$108$		0			0
$109$	−156.991			−1.44028			−0.720141	−	0.693828i	$-0.755923\pi$
$109$	−156.991			−1.44028			−0.720141	+	0.693828i	$0.755923\pi$
$110$		0			0
$111$	43.3019			0.390108
$112$		0			0
$113$		−	187.859i		−	1.66247i	−0.555924	−	0.831233i	$-0.687635\pi$
$113$		−	187.859i		−	1.66247i	0.555924	−	0.831233i	$-0.312365\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−143.581			−1.22719
$118$		0			0
$119$		−	176.224i		−	1.48088i
$120$		0			0
$121$	−76.0379			−0.628412
$122$		0			0
$123$	−74.4824			−0.605548
$124$		0			0
$125$		0			0
$126$		0			0
$127$	173.993			1.37002			0.685012	−	0.728532i	$-0.259797\pi$
$127$	173.993			1.37002			0.685012	+	0.728532i	$0.259797\pi$
$128$		0			0
$129$	1.56602			0.0121397
$130$		0			0
$131$	−130.868			−0.998992			−0.499496	−	0.866316i	$-0.666481\pi$
$131$	−130.868			−0.998992			−0.499496	+	0.866316i	$0.666481\pi$
$132$		0			0
$133$			78.6510i			0.591361i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−46.4084			−0.338747			−0.169374	−	0.985552i	$-0.554174\pi$
$137$	−46.4084			−0.338747			−0.169374	+	0.985552i	$0.554174\pi$
$138$		0			0
$139$		−	138.517i		−	0.996523i	−0.867027	−	0.498262i	$-0.833972\pi$
$139$		−	138.517i		−	0.996523i	0.867027	−	0.498262i	$-0.166028\pi$
$140$		0			0
$141$			27.7436i			0.196763i
$142$		0			0
$143$		−	271.114i		−	1.89590i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	12.9508			0.0881010
$148$		0			0
$149$	−167.548			−1.12448			−0.562240	−	0.826974i	$-0.690060\pi$
$149$	−167.548			−1.12448			−0.562240	+	0.826974i	$0.690060\pi$
$150$		0			0
$151$		−	179.270i		−	1.18722i	−0.804753	−	0.593610i	$-0.797702\pi$
$151$		−	179.270i		−	1.18722i	0.804753	−	0.593610i	$-0.202298\pi$
$152$		0			0
$153$	−210.720			−1.37726
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	44.3112i		−	0.282237i	−0.989993	−	0.141118i	$-0.954930\pi$
$157$		−	44.3112i		−	0.282237i	0.989993	−	0.141118i	$-0.0450698\pi$
$158$		0			0
$159$	−39.4903			−0.248367
$160$		0			0
$161$		−	32.8081i		−	0.203777i
$162$		0			0
$163$		−	270.821i		−	1.66148i	−0.556662	−	0.830739i	$-0.687918\pi$
$163$		−	270.821i		−	1.66148i	0.556662	−	0.830739i	$-0.312082\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	160.936			0.963687			0.481844	−	0.876257i	$-0.339967\pi$
$167$	160.936			0.963687			0.481844	+	0.876257i	$0.339967\pi$
$168$		0			0
$169$	204.038			1.20732
$170$		0			0
$171$	94.0471			0.549983
$172$		0			0
$173$			74.3398i			0.429710i	0.976646	+	0.214855i	$0.0689279\pi$
$173$			74.3398i			0.429710i	−0.976646	+	0.214855i	$0.931072\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	52.5002			0.296611
$178$		0			0
$179$		−	24.8629i		−	0.138899i	−0.997585	−	0.0694494i	$-0.977876\pi$
$179$		−	24.8629i		−	0.138899i	0.997585	−	0.0694494i	$-0.0221243\pi$
$180$		0			0
$181$		−	93.5250i		−	0.516713i	−0.966050	−	0.258356i	$-0.916819\pi$
$181$		−	93.5250i		−	0.516713i	0.966050	−	0.258356i	$-0.0831809\pi$
$182$		0			0
$183$		−	58.6223i		−	0.320341i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	397.887i		−	2.12774i
$188$		0			0
$189$			127.856i			0.676487i
$190$		0			0
$191$	102.557			0.536947			0.268473	−	0.963287i	$-0.413481\pi$
$191$	102.557			0.536947			0.268473	+	0.963287i	$0.413481\pi$
$192$		0			0
$193$		−	120.557i		−	0.624647i	−0.949976	−	0.312323i	$-0.898893\pi$
$193$		−	120.557i		−	0.624647i	0.949976	−	0.312323i	$-0.101107\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−214.475			−1.08870			−0.544352	−	0.838857i	$-0.683224\pi$
$197$	−214.475			−1.08870			−0.544352	+	0.838857i	$0.683224\pi$
$198$		0			0
$199$			280.410i			1.40909i	0.709657	+	0.704547i	$0.248850\pi$
$199$			280.410i			1.40909i	−0.709657	+	0.704547i	$0.751150\pi$
$200$		0			0
$201$		−	56.4785i		−	0.280988i
$202$		0			0
$203$	−213.436			−1.05141
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−39.2304			−0.189519
$208$		0			0
$209$			177.582i			0.849674i
$210$		0			0
$211$	284.991			1.35067			0.675334	−	0.737512i	$-0.264000\pi$
$211$	284.991			1.35067			0.675334	+	0.737512i	$0.264000\pi$
$212$		0			0
$213$	84.1632			0.395132
$214$		0			0
$215$		0			0
$216$		0			0
$217$	126.168	−	145.689i	0.581419	−	0.671377i
$218$		0			0
$219$	−60.8680			−0.277936
$220$		0			0
$221$	547.472			2.47725
$222$		0			0
$223$	−375.729			−1.68489			−0.842443	−	0.538786i	$-0.818883\pi$
$223$	−375.729			−1.68489			−0.842443	+	0.538786i	$0.818883\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			288.416i			1.27055i	0.772285	+	0.635277i	$0.219114\pi$
$227$			288.416i			1.27055i	−0.772285	+	0.635277i	$0.780886\pi$
$228$		0			0
$229$			188.903i			0.824906i	0.910979	+	0.412453i	$0.135328\pi$
$229$			188.903i			0.824906i	−0.910979	+	0.412453i	$0.864672\pi$
$230$		0			0
$231$	−109.208			−0.472761
$232$		0			0
$233$			225.689i			0.968622i	0.874896	+	0.484311i	$0.160930\pi$
$233$			225.689i			0.968622i	−0.874896	+	0.484311i	$0.839070\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			189.208i			0.798345i
$238$		0			0
$239$			29.4318i			0.123145i	0.998103	+	0.0615727i	$0.0196116\pi$
$239$			29.4318i			0.123145i	−0.998103	+	0.0615727i	$0.980388\pi$
$240$		0			0
$241$			199.564i			0.828067i	0.910262	+	0.414034i	$0.135880\pi$
$241$			199.564i			0.828067i	−0.910262	+	0.414034i	$0.864120\pi$
$242$		0			0
$243$	236.611			0.973706
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−244.343			−0.989244
$248$		0			0
$249$	−109.754			−0.440781
$250$		0			0
$251$		−	386.449i		−	1.53964i	−0.638263	−	0.769819i	$-0.720347\pi$
$251$		−	386.449i		−	1.53964i	0.638263	−	0.769819i	$-0.279653\pi$
$252$		0			0
$253$		−	74.0757i		−	0.292789i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	263.783i		−	1.02639i	−0.858271	−	0.513197i	$-0.828461\pi$
$257$		−	263.783i		−	1.02639i	0.858271	−	0.513197i	$-0.171539\pi$
$258$		0			0
$259$		−	215.124i		−	0.830595i
$260$		0			0
$261$			255.217i			0.977841i
$262$		0			0
$263$	376.544			1.43173			0.715863	−	0.698241i	$-0.246033\pi$
$263$	376.544			1.43173			0.715863	+	0.698241i	$0.246033\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	122.755i		−	0.459758i
$268$		0			0
$269$		−	194.665i		−	0.723661i	−0.932244	−	0.361831i	$-0.882152\pi$
$269$		−	194.665i		−	0.723661i	0.932244	−	0.361831i	$-0.117848\pi$
$270$		0			0
$271$			147.820i			0.545460i	0.962091	+	0.272730i	$0.0879266\pi$
$271$			147.820i			0.545460i	−0.962091	+	0.272730i	$0.912073\pi$
$272$		0			0
$273$		−	150.264i		−	0.550418i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−179.164			−0.646801			−0.323400	−	0.946262i	$-0.604826\pi$
$277$	−179.164			−0.646801			−0.323400	+	0.946262i	$0.604826\pi$
$278$		0			0
$279$	−174.208	−	150.866i	−0.624401	−	0.540737i
$280$		0			0
$281$	13.7073			0.0487803			0.0243902	−	0.999703i	$-0.492236\pi$
$281$	13.7073			0.0487803			0.0243902	+	0.999703i	$0.492236\pi$
$282$		0			0
$283$			99.7359i			0.352424i	0.984352	+	0.176212i	$0.0563844\pi$
$283$			99.7359i			0.352424i	−0.984352	+	0.176212i	$0.943616\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			370.029i			1.28930i
$288$		0			0
$289$	514.472			1.78018
$290$		0			0
$291$		−	126.168i		−	0.433567i
$292$		0			0
$293$		−	567.359i		−	1.93638i	−0.250217	−	0.968190i	$-0.580502\pi$
$293$		−	567.359i		−	1.93638i	0.250217	−	0.968190i	$-0.419498\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			288.680i			0.971985i
$298$		0			0
$299$	101.924			0.340884
$300$		0			0
$301$		−	7.77998i		−	0.0258471i
$302$		0			0
$303$	−22.0412			−0.0727431
$304$		0			0
$305$		0			0
$306$		0			0
$307$			311.877i			1.01589i	0.861391	+	0.507943i	$0.169594\pi$
$307$			311.877i			1.01589i	−0.861391	+	0.507943i	$0.830406\pi$
$308$		0			0
$309$		−	52.0749i		−	0.168527i
$310$		0			0
$311$	560.802			1.80322			0.901612	−	0.432546i	$-0.142385\pi$
$311$	560.802			1.80322			0.901612	+	0.432546i	$0.142385\pi$
$312$		0			0
$313$	−546.192			−1.74502			−0.872512	−	0.488593i	$-0.837510\pi$
$313$	−546.192			−1.74502			−0.872512	+	0.488593i	$0.837510\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	57.8587i		−	0.182520i	−0.995827	−	0.0912599i	$-0.970911\pi$
$317$		−	57.8587i		−	0.182520i	0.995827	−	0.0912599i	$-0.0290894\pi$
$318$		0			0
$319$	−481.906			−1.51068
$320$		0			0
$321$			114.315i			0.356121i
$322$		0			0
$323$	−358.599			−1.11021
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−196.459			−0.600793
$328$		0			0
$329$	137.830			0.418936
$330$		0			0
$331$		−	597.500i		−	1.80514i	−0.430547	−	0.902568i	$-0.641680\pi$
$331$		−	597.500i		−	1.80514i	0.430547	−	0.902568i	$-0.358320\pi$
$332$		0			0
$333$	−257.235			−0.772478
$334$		0			0
$335$		0			0
$336$		0			0
$337$	16.1620			0.0479584			0.0239792	−	0.999712i	$-0.492366\pi$
$337$	16.1620			0.0479584			0.0239792	+	0.999712i	$0.492366\pi$
$338$		0			0
$339$		−	235.088i		−	0.693474i
$340$		0			0
$341$	284.868	−	328.943i	0.835390	−	0.964643i
$342$		0			0
$343$		−	368.972i		−	1.07572i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−437.379			−1.26046			−0.630229	−	0.776409i	$-0.717039\pi$
$347$	−437.379			−1.26046			−0.630229	+	0.776409i	$0.717039\pi$
$348$		0			0
$349$	466.528			1.33676			0.668378	−	0.743821i	$-0.266989\pi$
$349$	466.528			1.33676			0.668378	+	0.743821i	$0.266989\pi$
$350$		0			0
$351$	−397.208			−1.13165
$352$		0			0
$353$	−64.4122			−0.182471			−0.0912354	−	0.995829i	$-0.529082\pi$
$353$	−64.4122			−0.182471			−0.0912354	+	0.995829i	$0.529082\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	220.528i		−	0.617726i
$358$		0			0
$359$	−545.312			−1.51898			−0.759488	−	0.650522i	$-0.774550\pi$
$359$	−545.312			−1.51898			−0.759488	+	0.650522i	$0.774550\pi$
$360$		0			0
$361$	−200.953			−0.556656
$362$		0			0
$363$	−95.1543			−0.262133
$364$		0			0
$365$		0			0
$366$		0			0
$367$	266.810			0.727002			0.363501	−	0.931594i	$-0.381581\pi$
$367$	266.810			0.727002			0.363501	+	0.931594i	$0.381581\pi$
$368$		0			0
$369$	442.463			1.19909
$370$		0			0
$371$			196.188i			0.528808i
$372$		0			0
$373$			142.727i			0.382645i	0.981527	+	0.191323i	$0.0612777\pi$
$373$			142.727i			0.382645i	−0.981527	+	0.191323i	$0.938722\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	663.077i		−	1.75882i
$378$		0			0
$379$	269.284			0.710511			0.355255	−	0.934769i	$-0.384394\pi$
$379$	269.284			0.710511			0.355255	+	0.934769i	$0.384394\pi$
$380$		0			0
$381$	217.736			0.571485
$382$		0			0
$383$	231.770			0.605144			0.302572	−	0.953126i	$-0.402155\pi$
$383$	231.770			0.605144			0.302572	+	0.953126i	$0.402155\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−9.30293			−0.0240386
$388$		0			0
$389$			310.337i			0.797782i	0.916998	+	0.398891i	$0.130605\pi$
$389$			310.337i			0.797782i	−0.916998	+	0.398891i	$0.869395\pi$
$390$		0			0
$391$	149.584			0.382569
$392$		0			0
$393$	−163.769			−0.416715
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	58.9150i		−	0.148401i	−0.997243	−	0.0742003i	$-0.976360\pi$
$397$		−	58.9150i		−	0.148401i	0.997243	−	0.0742003i	$-0.0236404\pi$
$398$		0			0
$399$			98.4244i			0.246678i
$400$		0			0
$401$		−	534.764i		−	1.33358i	−0.745247	−	0.666789i	$-0.767668\pi$
$401$		−	534.764i		−	1.33358i	0.745247	−	0.666789i	$-0.232332\pi$
$402$		0			0
$403$	452.608	+	391.963i	1.12310	+	0.972613i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	485.717i		−	1.19341i
$408$		0			0
$409$			409.954i			1.00233i	0.865351	+	0.501166i	$0.167096\pi$
$409$			409.954i			1.00233i	−0.865351	+	0.501166i	$0.832904\pi$
$410$		0			0
$411$	−58.0757			−0.141303
$412$		0			0
$413$		−	260.821i		−	0.631528i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	173.341i		−	0.415685i
$418$		0			0
$419$	−147.255			−0.351444			−0.175722	−	0.984440i	$-0.556226\pi$
$419$	−147.255			−0.351444			−0.175722	+	0.984440i	$0.556226\pi$
$420$		0			0
$421$	−86.8209			−0.206225			−0.103113	−	0.994670i	$-0.532880\pi$
$421$	−86.8209			−0.206225			−0.103113	+	0.994670i	$0.532880\pi$
$422$		0			0
$423$		−	164.811i		−	0.389623i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−291.236			−0.682051
$428$		0			0
$429$		−	339.273i		−	0.790847i
$430$		0			0
$431$	121.830			0.282668			0.141334	−	0.989962i	$-0.454861\pi$
$431$	121.830			0.282668			0.141334	+	0.989962i	$0.454861\pi$
$432$		0			0
$433$	−630.957			−1.45718			−0.728588	−	0.684952i	$-0.759823\pi$
$433$	−630.957			−1.45718			−0.728588	+	0.684952i	$0.759823\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−66.7613			−0.152772
$438$		0			0
$439$	−181.537			−0.413525			−0.206762	−	0.978391i	$-0.566293\pi$
$439$	−181.537			−0.413525			−0.206762	+	0.978391i	$0.566293\pi$
$440$		0			0
$441$	−76.9345			−0.174455
$442$		0			0
$443$			557.331i			1.25808i	0.777372	+	0.629041i	$0.216552\pi$
$443$			557.331i			1.25808i	−0.777372	+	0.629041i	$0.783448\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−209.670			−0.469061
$448$		0			0
$449$		−	258.428i		−	0.575563i	−0.957696	−	0.287781i	$-0.907082\pi$
$449$		−	258.428i		−	0.575563i	0.957696	−	0.287781i	$-0.0929176\pi$
$450$		0			0
$451$			835.468i			1.85248i
$452$		0			0
$453$		−	224.340i		−	0.495231i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−370.936			−0.811677			−0.405839	−	0.913945i	$-0.633020\pi$
$457$	−370.936			−0.811677			−0.405839	+	0.913945i	$0.633020\pi$
$458$		0			0
$459$	−582.944			−1.27003
$460$		0			0
$461$		−	222.078i		−	0.481731i	−0.970559	−	0.240865i	$-0.922569\pi$
$461$		−	222.078i		−	0.481731i	0.970559	−	0.240865i	$-0.0774313\pi$
$462$		0			0
$463$	69.5357			0.150185			0.0750925	−	0.997177i	$-0.476075\pi$
$463$	69.5357			0.150185			0.0750925	+	0.997177i	$0.476075\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			137.273i			0.293947i	0.989140	+	0.146974i	$0.0469532\pi$
$467$			137.273i			0.293947i	−0.989140	+	0.146974i	$0.953047\pi$
$468$		0			0
$469$	−280.585			−0.598263
$470$		0			0
$471$		−	55.4513i		−	0.117731i
$472$		0			0
$473$		−	17.5660i		−	0.0371375i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	234.592			0.491807
$478$		0			0
$479$	795.652			1.66107			0.830534	−	0.556967i	$-0.188035\pi$
$479$	795.652			1.66107			0.830534	+	0.556967i	$0.188035\pi$
$480$		0			0
$481$	668.321			1.38944
$482$		0			0
$483$		−	41.0563i		−	0.0850027i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	355.435			0.729847			0.364923	−	0.931038i	$-0.381095\pi$
$487$	355.435			0.729847			0.364923	+	0.931038i	$0.381095\pi$
$488$		0			0
$489$		−	338.907i		−	0.693062i
$490$		0			0
$491$		−	425.018i		−	0.865618i	−0.901486	−	0.432809i	$-0.857522\pi$
$491$		−	425.018i		−	0.865618i	0.901486	−	0.432809i	$-0.142478\pi$
$492$		0			0
$493$		−	973.134i		−	1.97390i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	418.123i		−	0.841293i
$498$		0			0
$499$			615.775i			1.23402i	0.786956	+	0.617009i	$0.211656\pi$
$499$			615.775i			1.23402i	−0.786956	+	0.617009i	$0.788344\pi$
$500$		0			0
$501$	201.396			0.401988
$502$		0			0
$503$		−	613.048i		−	1.21878i	−0.792869	−	0.609392i	$-0.791414\pi$
$503$		−	613.048i		−	1.21878i	0.792869	−	0.609392i	$-0.208586\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	255.334			0.503618
$508$		0			0
$509$			245.418i			0.482157i	0.970506	+	0.241078i	$0.0775011\pi$
$509$			245.418i			0.482157i	−0.970506	+	0.241078i	$0.922499\pi$
$510$		0			0
$511$			302.392i			0.591765i
$512$		0			0
$513$	260.175			0.507163
$514$		0			0
$515$		0			0
$516$		0			0
$517$	311.199			0.601933
$518$		0			0
$519$			93.0293i			0.179247i
$520$		0			0
$521$	49.6602			0.0953171			0.0476585	−	0.998864i	$-0.484824\pi$
$521$	49.6602			0.0953171			0.0476585	+	0.998864i	$0.484824\pi$
$522$		0			0
$523$	−114.421			−0.218778			−0.109389	−	0.993999i	$-0.534889\pi$
$523$	−114.421			−0.218778			−0.109389	+	0.993999i	$0.534889\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	664.250	+	575.247i	1.26044	+	1.09155i
$528$		0			0
$529$	−501.151			−0.947356
$530$		0			0
$531$	−311.877			−0.587339
$532$		0			0
$533$	−1149.56			−2.15677
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	31.1136i		−	0.0579397i
$538$		0			0
$539$		−	145.269i		−	0.269517i
$540$		0			0
$541$	−360.933			−0.667160			−0.333580	−	0.942722i	$-0.608257\pi$
$541$	−360.933			−0.667160			−0.333580	+	0.942722i	$0.608257\pi$
$542$		0			0
$543$		−	117.038i		−	0.215539i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			855.803i			1.56454i	0.622939	+	0.782270i	$0.285938\pi$
$547$			855.803i			1.56454i	−0.622939	+	0.782270i	$0.714062\pi$
$548$		0			0
$549$			348.246i			0.634328i
$550$		0			0
$551$			434.321i			0.788242i
$552$		0			0
$553$	939.984			1.69979
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−541.565			−0.972288			−0.486144	−	0.873879i	$-0.661597\pi$
$557$	−541.565			−0.972288			−0.486144	+	0.873879i	$0.661597\pi$
$558$		0			0
$559$	24.1699			0.0432378
$560$		0			0
$561$		−	497.919i		−	0.887556i
$562$		0			0
$563$		−	854.104i		−	1.51706i	−0.651639	−	0.758530i	$-0.725918\pi$
$563$		−	854.104i		−	1.51706i	0.651639	−	0.758530i	$-0.274082\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	255.953i		−	0.451416i
$568$		0			0
$569$		−	105.874i		−	0.186070i	−0.995663	−	0.0930350i	$-0.970343\pi$
$569$		−	105.874i		−	0.186070i	0.995663	−	0.0930350i	$-0.0296569\pi$
$570$		0			0
$571$			712.878i			1.24847i	0.781236	+	0.624236i	$0.214590\pi$
$571$			712.878i			1.24847i	−0.781236	+	0.624236i	$0.785410\pi$
$572$		0			0
$573$	128.340			0.223980
$574$		0			0
$575$		0			0
$576$		0			0
$577$			626.624i			1.08600i	0.839731	+	0.543002i	$0.182712\pi$
$577$			626.624i			1.08600i	−0.839731	+	0.543002i	$0.817288\pi$
$578$		0			0
$579$		−	150.866i		−	0.260562i
$580$		0			0
$581$			545.260i			0.938485i
$582$		0			0
$583$			442.962i			0.759798i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	287.151			0.489184			0.244592	−	0.969626i	$-0.421346\pi$
$587$	287.151			0.489184			0.244592	+	0.969626i	$0.421346\pi$
$588$		0			0
$589$	−296.463	−	256.739i	−0.503332	−	0.435890i
$590$		0			0
$591$	−268.395			−0.454137
$592$		0			0
$593$			850.029i			1.43344i	0.697362	+	0.716719i	$0.254357\pi$
$593$			850.029i			1.43344i	−0.697362	+	0.716719i	$0.745643\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			350.907i			0.587784i
$598$		0			0
$599$	−89.0092			−0.148596			−0.0742982	−	0.997236i	$-0.523672\pi$
$599$	−89.0092			−0.148596			−0.0742982	+	0.997236i	$0.523672\pi$
$600$		0			0
$601$			60.2212i			0.100202i	0.998744	+	0.0501009i	$0.0159543\pi$
$601$			60.2212i			0.100202i	−0.998744	+	0.0501009i	$0.984046\pi$
$602$		0			0
$603$			335.511i			0.556403i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	22.9048i		−	0.0377345i	−0.999822	−	0.0188672i	$-0.993994\pi$
$607$		−	22.9048i		−	0.0377345i	0.999822	−	0.0188672i	$-0.00600599\pi$
$608$		0			0
$609$	−267.095			−0.438580
$610$		0			0
$611$			428.194i			0.700808i
$612$		0			0
$613$	330.029			0.538384			0.269192	−	0.963087i	$-0.413243\pi$
$613$	330.029			0.538384			0.269192	+	0.963087i	$0.413243\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	800.907i		−	1.29807i	−0.760760	−	0.649033i	$-0.775174\pi$
$617$		−	800.907i		−	1.29807i	0.760760	−	0.649033i	$-0.224826\pi$
$618$		0			0
$619$			1023.01i			1.65269i	0.563165	+	0.826344i	$0.309583\pi$
$619$			1023.01i			1.65269i	−0.563165	+	0.826344i	$0.690417\pi$
$620$		0			0
$621$	−108.528			−0.174764
$622$		0			0
$623$	−609.849			−0.978891
$624$		0			0
$625$		0			0
$626$		0			0
$627$			222.227i			0.354429i
$628$		0			0
$629$	980.831			1.55935
$630$		0			0
$631$		−	111.305i		−	0.176394i	−0.996103	−	0.0881971i	$-0.971889\pi$
$631$		−	111.305i		−	0.176394i	0.996103	−	0.0881971i	$-0.0281105\pi$
$632$		0			0
$633$	356.639			0.563411
$634$		0			0
$635$		0			0
$636$		0			0
$637$	199.883			0.313788
$638$		0			0
$639$	−499.971			−0.782428
$640$		0			0
$641$		−	1219.04i		−	1.90177i	−0.309539	−	0.950887i	$-0.600175\pi$
$641$		−	1219.04i		−	1.90177i	0.309539	−	0.950887i	$-0.399825\pi$
$642$		0			0
$643$	523.171			0.813641			0.406821	−	0.913508i	$-0.366637\pi$
$643$	523.171			0.813641			0.406821	+	0.913508i	$0.366637\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−408.691			−0.631671			−0.315836	−	0.948814i	$-0.602285\pi$
$647$	−408.691			−0.631671			−0.315836	+	0.948814i	$0.602285\pi$
$648$		0			0
$649$		−	588.894i		−	0.907386i
$650$		0			0
$651$	157.887	−	182.316i	0.242531	−	0.280055i
$652$		0			0
$653$			394.340i			0.603889i	0.953325	+	0.301945i	$0.0976358\pi$
$653$			394.340i			0.603889i	−0.953325	+	0.301945i	$0.902364\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	361.586			0.550359
$658$		0			0
$659$	561.539			0.852108			0.426054	−	0.904698i	$-0.359903\pi$
$659$	561.539			0.852108			0.426054	+	0.904698i	$0.359903\pi$
$660$		0			0
$661$	538.162			0.814163			0.407081	−	0.913392i	$-0.366547\pi$
$661$	538.162			0.814163			0.407081	+	0.913392i	$0.366547\pi$
$662$		0			0
$663$	685.110			1.03335
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	181.171i		−	0.271621i
$668$		0			0
$669$	−470.190			−0.702826
$670$		0			0
$671$	−657.566			−0.979979
$672$		0			0
$673$	−30.4001			−0.0451710			−0.0225855	−	0.999745i	$-0.507190\pi$
$673$	−30.4001			−0.0451710			−0.0225855	+	0.999745i	$0.507190\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	374.537			0.553230			0.276615	−	0.960981i	$-0.410787\pi$
$677$	374.537			0.553230			0.276615	+	0.960981i	$0.410787\pi$
$678$		0			0
$679$	−626.802			−0.923126
$680$		0			0
$681$			360.925i			0.529993i
$682$		0			0
$683$			704.839i			1.03198i	0.856596	+	0.515988i	$0.172575\pi$
$683$			704.839i			1.03198i	−0.856596	+	0.515988i	$0.827425\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			236.395i			0.344098i
$688$		0			0
$689$	−609.492			−0.884604
$690$		0			0
$691$	320.160			0.463328			0.231664	−	0.972796i	$-0.425583\pi$
$691$	320.160			0.463328			0.231664	+	0.972796i	$0.425583\pi$
$692$		0			0
$693$	648.749			0.936145
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1687.10			−2.42051
$698$		0			0
$699$			282.429i			0.404047i
$700$		0			0
$701$	−1132.37			−1.61536			−0.807681	−	0.589620i	$-0.799278\pi$
$701$	−1132.37			−1.61536			−0.807681	+	0.589620i	$0.799278\pi$
$702$		0			0
$703$	−437.757			−0.622698
$704$		0			0
$705$		0			0
$706$		0			0
$707$			109.500i			0.154880i
$708$		0			0
$709$		−	1096.08i		−	1.54595i	−0.634435	−	0.772976i	$-0.718767\pi$
$709$		−	1096.08i		−	1.54595i	0.634435	−	0.772976i	$-0.281233\pi$
$710$		0			0
$711$		−	1123.99i		−	1.58086i
$712$		0			0
$713$	123.665	+	107.095i	0.173443	+	0.150204i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			36.8311i			0.0513683i
$718$		0			0
$719$			724.199i			1.00723i	0.863928	+	0.503616i	$0.167997\pi$
$719$			724.199i			1.00723i	−0.863928	+	0.503616i	$0.832003\pi$
$720$		0			0
$721$	−258.708			−0.358819
$722$		0			0
$723$			249.736i			0.345416i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	383.783i		−	0.527900i	−0.964536	−	0.263950i	$-0.914975\pi$
$727$		−	383.783i		−	0.527900i	0.964536	−	0.263950i	$-0.0850253\pi$
$728$		0			0
$729$	−74.4330			−0.102103
$730$		0			0
$731$	35.4718			0.0485251
$732$		0			0
$733$			1217.45i			1.66091i	0.557088	+	0.830454i	$0.311919\pi$
$733$			1217.45i			1.66091i	−0.557088	+	0.830454i	$0.688081\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−633.519			−0.859592
$738$		0			0
$739$		−	1097.11i		−	1.48458i	−0.670077	−	0.742292i	$-0.733739\pi$
$739$		−	1097.11i		−	1.48458i	0.670077	−	0.742292i	$-0.266261\pi$
$740$		0			0
$741$	−305.773			−0.412649
$742$		0			0
$743$	−800.583			−1.07750			−0.538750	−	0.842466i	$-0.681103\pi$
$743$	−800.583			−1.07750			−0.538750	+	0.842466i	$0.681103\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	651.996			0.872819
$748$		0			0
$749$	567.916			0.758232
$750$		0			0
$751$	410.708			0.546882			0.273441	−	0.961889i	$-0.411838\pi$
$751$	410.708			0.546882			0.273441	+	0.961889i	$0.411838\pi$
$752$		0			0
$753$		−	483.605i		−	0.642238i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−47.6008			−0.0628809			−0.0314405	−	0.999506i	$-0.510009\pi$
$757$	−47.6008			−0.0628809			−0.0314405	+	0.999506i	$0.510009\pi$
$758$		0			0
$759$		−	92.6989i		−	0.122133i
$760$		0			0
$761$			324.374i			0.426248i	0.977025	+	0.213124i	$0.0683638\pi$
$761$			324.374i			0.426248i	−0.977025	+	0.213124i	$0.931636\pi$
$762$		0			0
$763$			976.010i			1.27917i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	810.286			1.05644
$768$		0			0
$769$	793.067			1.03130			0.515648	−	0.856801i	$-0.327551\pi$
$769$	793.067			1.03130			0.515648	+	0.856801i	$0.327551\pi$
$770$		0			0
$771$		−	330.100i		−	0.428145i
$772$		0			0
$773$	−664.261			−0.859329			−0.429664	−	0.902989i	$-0.641368\pi$
$773$	−664.261			−0.859329			−0.429664	+	0.902989i	$0.641368\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		−	269.208i		−	0.346471i
$778$		0			0
$779$	752.972			0.966588
$780$		0			0
$781$		−	944.058i		−	1.20878i
$782$		0			0
$783$			706.039i			0.901710i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	845.550			1.07440			0.537198	−	0.843456i	$-0.319483\pi$
$787$	845.550			1.07440			0.537198	+	0.843456i	$0.319483\pi$
$788$		0			0
$789$	471.210			0.597224
$790$		0			0
$791$	−1167.92			−1.47651
$792$		0			0
$793$		−	904.776i		−	1.14095i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1166.58			1.46372			0.731859	−	0.681456i	$-0.238653\pi$
$797$	1166.58			1.46372			0.731859	+	0.681456i	$0.238653\pi$
$798$		0			0
$799$			628.419i			0.786507i
$800$		0			0
$801$			729.228i			0.910397i
$802$		0			0
$803$			682.755i			0.850256i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	243.605i		−	0.301865i
$808$		0			0
$809$		−	1425.02i		−	1.76146i	−0.473616	−	0.880731i	$-0.657052\pi$
$809$		−	1425.02i		−	1.76146i	0.473616	−	0.880731i	$-0.342948\pi$
$810$		0			0
$811$	−1041.77			−1.28456			−0.642278	−	0.766472i	$-0.722010\pi$
$811$	−1041.77			−1.28456			−0.642278	+	0.766472i	$0.722010\pi$
$812$		0			0
$813$			184.983i			0.227531i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−15.8315			−0.0193776
$818$		0			0
$819$			892.644i			1.08992i
$820$		0			0
$821$			1346.40i			1.63995i	0.572401	+	0.819974i	$0.306012\pi$
$821$			1346.40i			1.63995i	−0.572401	+	0.819974i	$0.693988\pi$
$822$		0			0
$823$	1067.20			1.29672			0.648361	−	0.761333i	$-0.275455\pi$
$823$	1067.20			1.29672			0.648361	+	0.761333i	$0.275455\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	429.658			0.519538			0.259769	−	0.965671i	$-0.416354\pi$
$827$	429.658			0.519538			0.259769	+	0.965671i	$0.416354\pi$
$828$		0			0
$829$		−	286.631i		−	0.345755i	−0.984943	−	0.172878i	$-0.944693\pi$
$829$		−	286.631i		−	0.345755i	0.984943	−	0.172878i	$-0.0553065\pi$
$830$		0			0
$831$	−224.207			−0.269804
$832$		0			0
$833$	293.349			0.352160
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−481.934	−	417.359i	−0.575787	−	0.498637i
$838$		0			0
$839$	14.8864			0.0177430			0.00887151	−	0.999961i	$-0.497176\pi$
$839$	14.8864			0.0177430			0.00887151	+	0.999961i	$0.497176\pi$
$840$		0			0
$841$	−337.622			−0.401453
$842$		0			0
$843$	17.1534			0.0203480
$844$		0			0
$845$		0			0
$846$		0			0
$847$			472.727i			0.558119i
$848$		0			0
$849$			124.810i			0.147008i
$850$		0			0
$851$	182.604			0.214576
$852$		0			0
$853$		−	411.171i		−	0.482029i	−0.970521	−	0.241015i	$-0.922520\pi$
$853$		−	411.171i		−	0.482029i	0.970521	−	0.241015i	$-0.0774802\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			252.227i			0.294314i	0.989113	+	0.147157i	$0.0470123\pi$
$857$			252.227i			0.294314i	−0.989113	+	0.147157i	$0.952988\pi$
$858$		0			0
$859$			657.355i			0.765256i	0.923903	+	0.382628i	$0.124981\pi$
$859$			657.355i			0.765256i	−0.923903	+	0.382628i	$0.875019\pi$
$860$		0			0
$861$			463.056i			0.537812i
$862$		0			0
$863$	292.322			0.338728			0.169364	−	0.985554i	$-0.445829\pi$
$863$	292.322			0.338728			0.169364	+	0.985554i	$0.445829\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	643.814			0.742576
$868$		0			0
$869$	2122.34			2.44228
$870$		0			0
$871$		−	871.689i		−	1.00079i
$872$		0			0
$873$			749.500i			0.858534i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			649.236i			0.740292i	0.928974	+	0.370146i	$0.120692\pi$
$877$			649.236i			0.740292i	−0.928974	+	0.370146i	$0.879308\pi$
$878$		0			0
$879$		−	709.997i		−	0.807733i
$880$		0			0
$881$			455.276i			0.516772i	0.966042	+	0.258386i	$0.0831907\pi$
$881$			455.276i			0.516772i	−0.966042	+	0.258386i	$0.916809\pi$
$882$		0			0
$883$	29.2896			0.0331705			0.0165853	−	0.999862i	$-0.494721\pi$
$883$	29.2896			0.0331705			0.0165853	+	0.999862i	$0.494721\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	240.706i		−	0.271371i	−0.990752	−	0.135686i	$-0.956676\pi$
$887$		−	240.706i		−	0.271371i	0.990752	−	0.135686i	$-0.0433237\pi$
$888$		0			0
$889$		−	1081.71i		−	1.21677i
$890$		0			0
$891$		−	577.903i		−	0.648600i
$892$		0			0
$893$		−	280.471i		−	0.314077i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	127.549			0.142195
$898$		0			0
$899$	696.716	−	804.514i	0.774991	−	0.894898i
$900$		0			0
$901$	−894.493			−0.992778
$902$		0			0
$903$		−	9.73592i		−	0.0107818i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	1582.96i		−	1.74527i	−0.488377	−	0.872633i	$-0.662411\pi$
$907$		−	1582.96i		−	1.74527i	0.488377	−	0.872633i	$-0.337589\pi$
$908$		0			0
$909$	130.935			0.144043
$910$		0			0
$911$			1239.66i			1.36077i	0.732855	+	0.680385i	$0.238187\pi$
$911$			1239.66i			1.36077i	−0.732855	+	0.680385i	$0.761813\pi$
$912$		0			0
$913$			1231.11i			1.34843i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			813.605i			0.887246i
$918$		0			0
$919$	−1024.49			−1.11479			−0.557396	−	0.830247i	$-0.688199\pi$
$919$	−1024.49			−1.11479			−0.557396	+	0.830247i	$0.688199\pi$
$920$		0			0
$921$			390.285i			0.423762i
$922$		0			0
$923$	1298.97			1.40734
$924$		0			0
$925$		0			0
$926$		0			0
$927$			309.351i			0.333712i
$928$		0			0
$929$			1227.18i			1.32097i	0.750838	+	0.660486i	$0.229650\pi$
$929$			1227.18i			1.32097i	−0.750838	+	0.660486i	$0.770350\pi$
$930$		0			0
$931$	−130.925			−0.140629
$932$		0			0
$933$	701.792			0.752189
$934$		0			0
$935$		0			0
$936$		0			0
$937$			170.115i			0.181552i	0.995871	+	0.0907762i	$0.0289348\pi$
$937$			170.115i			0.181552i	−0.995871	+	0.0907762i	$0.971065\pi$
$938$		0			0
$939$	−683.509			−0.727911
$940$		0			0
$941$			1179.64i			1.25360i	0.779179	+	0.626802i	$0.215636\pi$
$941$			1179.64i			1.25360i	−0.779179	+	0.626802i	$0.784364\pi$
$942$		0			0
$943$	−314.092			−0.333077
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1558.86			−1.64611			−0.823054	−	0.567963i	$-0.807732\pi$
$947$	−1558.86			−1.64611			−0.823054	+	0.567963i	$0.807732\pi$
$948$		0			0
$949$	−939.435			−0.989921
$950$		0			0
$951$		−	72.4048i		−	0.0761355i
$952$		0			0
$953$	1624.65			1.70477			0.852385	−	0.522914i	$-0.175155\pi$
$953$	1624.65			1.70477			0.852385	+	0.522914i	$0.175155\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−603.060			−0.630157
$958$		0			0
$959$			288.520i			0.300855i
$960$		0			0
$961$	137.303	+	951.141i	0.142875	+	0.989741i
$962$		0			0
$963$		−	679.087i		−	0.705179i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	821.750			0.849793			0.424897	−	0.905242i	$-0.360310\pi$
$967$	821.750			0.849793			0.424897	+	0.905242i	$0.360310\pi$
$968$		0			0
$969$	−448.753			−0.463110
$970$		0			0
$971$	1287.55			1.32600			0.663001	−	0.748619i	$-0.269283\pi$
$971$	1287.55			1.32600			0.663001	+	0.748619i	$0.269283\pi$
$972$		0			0
$973$	−861.157			−0.885054
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	365.367i		−	0.373969i	−0.982363	−	0.186984i	$-0.940129\pi$
$977$		−	365.367i		−	0.373969i	0.982363	−	0.186984i	$-0.0598714\pi$
$978$		0			0
$979$	−1376.95			−1.40648
$980$		0			0
$981$	1167.07			1.18967
$982$		0			0
$983$	1602.82			1.63054			0.815268	−	0.579083i	$-0.196589\pi$
$983$	1602.82			1.63054			0.815268	+	0.579083i	$0.196589\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	172.482			0.174753
$988$		0			0
$989$	6.60389			0.00667734
$990$		0			0
$991$		−	86.9017i		−	0.0876909i	−0.999038	−	0.0438455i	$-0.986039\pi$
$991$		−	86.9017i		−	0.0876909i	0.999038	−	0.0438455i	$-0.0139609\pi$
$992$		0			0
$993$		−	747.715i		−	0.752986i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	1535.33i		−	1.53995i	−0.638073	−	0.769976i	$-0.720268\pi$
$997$		−	1535.33i		−	1.53995i	0.638073	−	0.769976i	$-0.279732\pi$
$998$		0			0
$999$	−711.623			−0.712336

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3100.3.f.b.1549.5		8
5.2	odd	4		124.3.c.b.61.2	✓	4
5.3	odd	4		3100.3.d.b.1301.3		4
5.4	even	2	inner	3100.3.f.b.1549.4		8
15.2	even	4		1116.3.h.d.433.4		4
20.7	even	4		496.3.e.e.433.3		4
31.30	odd	2	inner	3100.3.f.b.1549.3		8
155.92	even	4		124.3.c.b.61.3	yes	4
155.123	even	4		3100.3.d.b.1301.2		4
155.154	odd	2	inner	3100.3.f.b.1549.6		8
465.92	odd	4		1116.3.h.d.433.3		4
620.247	odd	4		496.3.e.e.433.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.3.c.b.61.2	✓	4	5.2	odd	4
124.3.c.b.61.3	yes	4	155.92	even	4
496.3.e.e.433.2		4	620.247	odd	4
496.3.e.e.433.3		4	20.7	even	4
1116.3.h.d.433.3		4	465.92	odd	4
1116.3.h.d.433.4		4	15.2	even	4
3100.3.d.b.1301.2		4	155.123	even	4
3100.3.d.b.1301.3		4	5.3	odd	4
3100.3.f.b.1549.3		8	31.30	odd	2	inner
3100.3.f.b.1549.4		8	5.4	even	2	inner
3100.3.f.b.1549.5		8	1.1	even	1	trivial
3100.3.f.b.1549.6		8	155.154	odd	2	inner

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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.25141 q^{3} -6.21699i q^{7} -7.43398 q^{9} +O(q^{10})\)	\(q+1.25141 q^{3} -6.21699i q^{7} -7.43398 q^{9} -14.0370i q^{11} +19.3142 q^{13} +28.3456 q^{17} -12.6510 q^{19} -7.77998i q^{21} +5.27717 q^{23} -20.5656 q^{27} -34.3311i q^{29} +(23.4340 + 20.2941i) q^{31} -17.5660i q^{33} +34.6026 q^{37} +24.1699 q^{39} -59.5189 q^{41} +1.25141 q^{43} +22.1699i q^{47} +10.3490 q^{49} +35.4718 q^{51} -31.5567 q^{53} -15.8315 q^{57} +41.9529 q^{59} -46.8451i q^{61} +46.2170i q^{63} -45.1320i q^{67} +6.60389 q^{69} +67.2549 q^{71} -48.6396 q^{73} -87.2680 q^{77} +151.196i q^{79} +41.1699 q^{81} -87.7048 q^{83} -42.9621i q^{87} -98.0939i q^{89} -120.076i q^{91} +(29.3254 + 25.3961i) q^{93} -100.821i q^{97} +104.351i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{9}+O(q^{10}) \) 8 * q + 16 * q^9	\( 8 q + 16 q^{9} + 12 q^{19} + 112 q^{31} - 184 q^{39} - 212 q^{41} + 196 q^{49} - 320 q^{51} - 4 q^{59} - 400 q^{69} - 28 q^{71} - 48 q^{81}+O(q^{100}) \) 8 * q + 16 * q^9 + 12 * q^19 + 112 * q^31 - 184 * q^39 - 212 * q^41 + 196 * q^49 - 320 * q^51 - 4 * q^59 - 400 * q^69 - 28 * q^71 - 48 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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