LMFDB - Embedded newform 3100.3.d.h.1301.12

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:	$24$
Twist minimal:	no (minimal twist has level 620)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1301.12
Character	$\chi$	$=$	3100.1301
Dual form			3100.3.d.h.1301.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.59213i q^{3} +1.60644 q^{7} +6.46514 q^{9} +O(q^{10})$	$q+1.59213i q^{3} +1.60644 q^{7} +6.46514 q^{9} -2.01438i q^{11} +1.88658i q^{13} -1.86366i q^{17} +6.62236 q^{19} +2.55765i q^{21} -27.0144i q^{23} +24.6224i q^{27} +45.2074i q^{29} +(-13.7468 + 27.7854i) q^{31} +3.20715 q^{33} +40.6209i q^{37} -3.00367 q^{39} -6.90007 q^{41} -47.4345i q^{43} +57.3430 q^{47} -46.4194 q^{49} +2.96718 q^{51} +0.0248289i q^{53} +10.5436i q^{57} +73.5797 q^{59} -81.6110i q^{61} +10.3858 q^{63} +50.0912 q^{67} +43.0104 q^{69} +77.8890 q^{71} -120.168i q^{73} -3.23598i q^{77} +106.078i q^{79} +18.9842 q^{81} +105.920i q^{83} -71.9758 q^{87} +76.7843i q^{89} +3.03066i q^{91} +(-44.2378 - 21.8866i) q^{93} +161.362 q^{97} -13.0233i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 44 q^{9}+O(q^{10}) $ 24 * q - 44 * q^9	$ 24 q - 44 q^{9} - 8 q^{19} - 24 q^{31} + 280 q^{39} - 248 q^{41} + 644 q^{49} - 100 q^{51} - 152 q^{59} + 288 q^{69} - 352 q^{71} - 368 q^{81}+O(q^{100}) $ 24 * q - 44 * q^9 - 8 * q^19 - 24 * q^31 + 280 * q^39 - 248 * q^41 + 644 * q^49 - 100 * q^51 - 152 * q^59 + 288 * q^69 - 352 * q^71 - 368 * 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
$3$			1.59213i			0.530709i	0.964151	+	0.265354i	$0.0854889\pi$
$3$			1.59213i			0.530709i	−0.964151	+	0.265354i	$0.914511\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.60644			0.229491			0.114745	−	0.993395i	$-0.463395\pi$
$7$	1.60644			0.229491			0.114745	+	0.993395i	$0.463395\pi$
$8$		0			0
$9$	6.46514			0.718348
$10$		0			0
$11$		−	2.01438i		−	0.183126i	−0.995799	−	0.0915630i	$-0.970814\pi$
$11$		−	2.01438i		−	0.183126i	0.995799	−	0.0915630i	$-0.0291863\pi$
$12$		0			0
$13$			1.88658i			0.145121i	0.997364	+	0.0725606i	$0.0231171\pi$
$13$			1.88658i			0.145121i	−0.997364	+	0.0725606i	$0.976883\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	1.86366i		−	0.109627i	−0.998497	−	0.0548135i	$-0.982544\pi$
$17$		−	1.86366i		−	0.109627i	0.998497	−	0.0548135i	$-0.0174564\pi$
$18$		0			0
$19$	6.62236			0.348545			0.174273	−	0.984697i	$-0.444243\pi$
$19$	6.62236			0.348545			0.174273	+	0.984697i	$0.444243\pi$
$20$		0			0
$21$			2.55765i			0.121793i
$22$		0			0
$23$		−	27.0144i		−	1.17454i	−0.809391	−	0.587270i	$-0.800203\pi$
$23$		−	27.0144i		−	1.17454i	0.809391	−	0.587270i	$-0.199797\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			24.6224i			0.911942i
$28$		0			0
$29$			45.2074i			1.55888i	0.626480	+	0.779438i	$0.284495\pi$
$29$			45.2074i			1.55888i	−0.626480	+	0.779438i	$0.715505\pi$
$30$		0			0
$31$	−13.7468	+	27.7854i	−0.443444	+	0.896302i
$31$	−13.7468	+	27.7854i	−0.443444	+	0.896302i
$32$		0			0
$33$	3.20715			0.0971865
$34$		0			0
$35$		0			0
$36$		0			0
$37$			40.6209i			1.09786i	0.835867	+	0.548932i	$0.184965\pi$
$37$			40.6209i			1.09786i	−0.835867	+	0.548932i	$0.815035\pi$
$38$		0			0
$39$	−3.00367			−0.0770171
$40$		0			0
$41$	−6.90007			−0.168294			−0.0841472	−	0.996453i	$-0.526817\pi$
$41$	−6.90007			−0.168294			−0.0841472	+	0.996453i	$0.526817\pi$
$42$		0			0
$43$		−	47.4345i		−	1.10313i	−0.834133	−	0.551564i	$-0.814031\pi$
$43$		−	47.4345i		−	1.10313i	0.834133	−	0.551564i	$-0.185969\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	57.3430			1.22006			0.610032	−	0.792376i	$-0.291156\pi$
$47$	57.3430			1.22006			0.610032	+	0.792376i	$0.291156\pi$
$48$		0			0
$49$	−46.4194			−0.947334
$50$		0			0
$51$	2.96718			0.0581800
$52$		0			0
$53$			0.0248289i			0.000468470i	1.00000		0.000234235i	$7.45593e-5\pi$
$53$			0.0248289i			0.000468470i	−1.00000		0.000234235i	$0.999925\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			10.5436i			0.184976i
$58$		0			0
$59$	73.5797			1.24711			0.623557	−	0.781778i	$-0.285687\pi$
$59$	73.5797			1.24711			0.623557	+	0.781778i	$0.285687\pi$
$60$		0			0
$61$		−	81.6110i		−	1.33788i	−0.743314	−	0.668942i	$-0.766747\pi$
$61$		−	81.6110i		−	1.33788i	0.743314	−	0.668942i	$-0.233253\pi$
$62$		0			0
$63$	10.3858			0.164854
$64$		0			0
$65$		0			0
$66$		0			0
$67$	50.0912			0.747630			0.373815	−	0.927503i	$-0.378049\pi$
$67$	50.0912			0.747630			0.373815	+	0.927503i	$0.378049\pi$
$68$		0			0
$69$	43.0104			0.623339
$70$		0			0
$71$	77.8890			1.09703			0.548514	−	0.836141i	$-0.315194\pi$
$71$	77.8890			1.09703			0.548514	+	0.836141i	$0.315194\pi$
$72$		0			0
$73$		−	120.168i		−	1.64614i	−0.567940	−	0.823070i	$-0.692259\pi$
$73$		−	120.168i		−	1.64614i	0.567940	−	0.823070i	$-0.307741\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	3.23598i		−	0.0420257i
$78$		0			0
$79$			106.078i			1.34276i	0.741115	+	0.671378i	$0.234297\pi$
$79$			106.078i			1.34276i	−0.741115	+	0.671378i	$0.765703\pi$
$80$		0			0
$81$	18.9842			0.234373
$82$		0			0
$83$			105.920i			1.27614i	0.769977	+	0.638072i	$0.220268\pi$
$83$			105.920i			1.27614i	−0.769977	+	0.638072i	$0.779732\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−71.9758			−0.827309
$88$		0			0
$89$			76.7843i			0.862745i	0.902174	+	0.431373i	$0.141971\pi$
$89$			76.7843i			0.862745i	−0.902174	+	0.431373i	$0.858029\pi$
$90$		0			0
$91$			3.03066i			0.0333040i
$92$		0			0
$93$	−44.2378	−	21.8866i	−0.475675	−	0.235339i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	161.362			1.66353			0.831764	−	0.555130i	$-0.187332\pi$
$97$	161.362			1.66353			0.831764	+	0.555130i	$0.187332\pi$
$98$		0			0
$99$		−	13.0233i		−	0.131548i
$100$		0			0
$101$	−126.932			−1.25675			−0.628374	−	0.777911i	$-0.716279\pi$
$101$	−126.932			−1.25675			−0.628374	+	0.777911i	$0.716279\pi$
$102$		0			0
$103$	35.2504			0.342237			0.171118	−	0.985250i	$-0.445262\pi$
$103$	35.2504			0.342237			0.171118	+	0.985250i	$0.445262\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−175.315			−1.63846			−0.819230	−	0.573465i	$-0.805599\pi$
$107$	−175.315			−1.63846			−0.819230	+	0.573465i	$0.805599\pi$
$108$		0			0
$109$	15.9398			0.146237			0.0731183	−	0.997323i	$-0.476705\pi$
$109$	15.9398			0.146237			0.0731183	+	0.997323i	$0.476705\pi$
$110$		0			0
$111$	−64.6736			−0.582645
$112$		0			0
$113$	56.1741			0.497116			0.248558	−	0.968617i	$-0.420043\pi$
$113$	56.1741			0.497116			0.248558	+	0.968617i	$0.420043\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			12.1970i			0.104248i
$118$		0			0
$119$		−	2.99385i		−	0.0251584i
$120$		0			0
$121$	116.942			0.966465
$122$		0			0
$123$		−	10.9858i		−	0.0893153i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	45.5941i		−	0.359009i	−0.983757	−	0.179504i	$-0.942551\pi$
$127$		−	45.5941i		−	0.359009i	0.983757	−	0.179504i	$-0.0574494\pi$
$128$		0			0
$129$	75.5217			0.585440
$130$		0			0
$131$	−44.3911			−0.338863			−0.169432	−	0.985542i	$-0.554193\pi$
$131$	−44.3911			−0.338863			−0.169432	+	0.985542i	$0.554193\pi$
$132$		0			0
$133$	10.6384			0.0799879
$134$		0			0
$135$		0			0
$136$		0			0
$137$			61.5738i			0.449444i	0.974423	+	0.224722i	$0.0721474\pi$
$137$			61.5738i			0.449444i	−0.974423	+	0.224722i	$0.927853\pi$
$138$		0			0
$139$			117.389i			0.844522i	0.906474	+	0.422261i	$0.138763\pi$
$139$			117.389i			0.844522i	−0.906474	+	0.422261i	$0.861237\pi$
$140$		0			0
$141$			91.2973i			0.647499i
$142$		0			0
$143$	3.80029			0.0265754
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	73.9055i		−	0.502758i
$148$		0			0
$149$	−54.8677			−0.368240			−0.184120	−	0.982904i	$-0.558943\pi$
$149$	−54.8677			−0.368240			−0.184120	+	0.982904i	$0.558943\pi$
$150$		0			0
$151$			232.960i			1.54278i	0.636362	+	0.771391i	$0.280439\pi$
$151$			232.960i			1.54278i	−0.636362	+	0.771391i	$0.719561\pi$
$152$		0			0
$153$		−	12.0488i		−	0.0787503i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	14.1768			0.0902978			0.0451489	−	0.998980i	$-0.485624\pi$
$157$	14.1768			0.0902978			0.0451489	+	0.998980i	$0.485624\pi$
$158$		0			0
$159$	−0.0395307			−0.000248621
$160$		0			0
$161$		−	43.3970i		−	0.269546i
$162$		0			0
$163$	90.5189			0.555331			0.277665	−	0.960678i	$-0.410439\pi$
$163$	90.5189			0.555331			0.277665	+	0.960678i	$0.410439\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	185.970i		−	1.11359i	−0.830649	−	0.556797i	$-0.812030\pi$
$167$		−	185.970i		−	1.11359i	0.830649	−	0.556797i	$-0.187970\pi$
$168$		0			0
$169$	165.441			0.978940
$170$		0			0
$171$	42.8144			0.250377
$172$		0			0
$173$	92.6355			0.535465			0.267733	−	0.963493i	$-0.413726\pi$
$173$	92.6355			0.535465			0.267733	+	0.963493i	$0.413726\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			117.148i			0.661854i
$178$		0			0
$179$		−	4.90300i		−	0.0273911i	−0.999906	−	0.0136955i	$-0.995640\pi$
$179$		−	4.90300i		−	0.0273911i	0.999906	−	0.0136955i	$-0.00435956\pi$
$180$		0			0
$181$			41.8556i			0.231246i	0.993293	+	0.115623i	$0.0368865\pi$
$181$			41.8556i			0.231246i	−0.993293	+	0.115623i	$0.963114\pi$
$182$		0			0
$183$	129.935			0.710027
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−3.75412			−0.0200755
$188$		0			0
$189$			39.5544i			0.209282i
$190$		0			0
$191$	226.964			1.18829			0.594145	−	0.804358i	$-0.297490\pi$
$191$	226.964			1.18829			0.594145	+	0.804358i	$0.297490\pi$
$192$		0			0
$193$	9.99542			0.0517898			0.0258949	−	0.999665i	$-0.491756\pi$
$193$	9.99542			0.0517898			0.0258949	+	0.999665i	$0.491756\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	269.638i		−	1.36872i	−0.729143	−	0.684362i	$-0.760081\pi$
$197$		−	269.638i		−	1.36872i	0.729143	−	0.684362i	$-0.239919\pi$
$198$		0			0
$199$			206.239i			1.03638i	0.855267	+	0.518188i	$0.173393\pi$
$199$			206.239i			1.03638i	−0.855267	+	0.518188i	$0.826607\pi$
$200$		0			0
$201$			79.7515i			0.396774i
$202$		0			0
$203$			72.6228i			0.357748i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	174.652i		−	0.843729i
$208$		0			0
$209$		−	13.3400i		−	0.0638276i
$210$		0			0
$211$	116.818			0.553640			0.276820	−	0.960922i	$-0.410719\pi$
$211$	116.818			0.553640			0.276820	+	0.960922i	$0.410719\pi$
$212$		0			0
$213$			124.009i			0.582202i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−22.0833	+	44.6354i	−0.101766	+	0.205693i
$218$		0			0
$219$	191.323			0.873621
$220$		0			0
$221$	3.51593			0.0159092
$222$		0			0
$223$			353.024i			1.58307i	0.611125	+	0.791534i	$0.290717\pi$
$223$			353.024i			1.58307i	−0.611125	+	0.791534i	$0.709283\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−219.707			−0.967873			−0.483937	−	0.875103i	$-0.660793\pi$
$227$	−219.707			−0.967873			−0.483937	+	0.875103i	$0.660793\pi$
$228$		0			0
$229$			291.138i			1.27135i	0.771958	+	0.635673i	$0.219277\pi$
$229$			291.138i			1.27135i	−0.771958	+	0.635673i	$0.780723\pi$
$230$		0			0
$231$	5.15209			0.0223034
$232$		0			0
$233$	−318.171			−1.36554			−0.682770	−	0.730633i	$-0.739225\pi$
$233$	−318.171			−1.36554			−0.682770	+	0.730633i	$0.739225\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−168.889			−0.712613
$238$		0			0
$239$			106.228i			0.444469i	0.974993	+	0.222234i	$0.0713350\pi$
$239$			106.228i			0.444469i	−0.974993	+	0.222234i	$0.928665\pi$
$240$		0			0
$241$		−	260.338i		−	1.08024i	−0.841588	−	0.540120i	$-0.818379\pi$
$241$		−	260.338i		−	1.08024i	0.841588	−	0.540120i	$-0.181621\pi$
$242$		0			0
$243$			251.827i			1.03633i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			12.4936i			0.0505813i
$248$		0			0
$249$	−168.638			−0.677260
$250$		0			0
$251$			481.362i			1.91778i	0.283784	+	0.958888i	$0.408410\pi$
$251$			481.362i			1.91778i	−0.283784	+	0.958888i	$0.591590\pi$
$252$		0			0
$253$	−54.4174			−0.215089
$254$		0			0
$255$		0			0
$256$		0			0
$257$	293.961			1.14382			0.571909	−	0.820317i	$-0.306203\pi$
$257$	293.961			1.14382			0.571909	+	0.820317i	$0.306203\pi$
$258$		0			0
$259$			65.2549i			0.251950i
$260$		0			0
$261$			292.272i			1.11982i
$262$		0			0
$263$		−	19.0346i		−	0.0723749i	−0.999345	−	0.0361874i	$-0.988479\pi$
$263$		−	19.0346i		−	0.0723749i	0.999345	−	0.0361874i	$-0.0115213\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−122.250			−0.457866
$268$		0			0
$269$			269.571i			1.00212i	0.865411	+	0.501062i	$0.167057\pi$
$269$			269.571i			1.00212i	−0.865411	+	0.501062i	$0.832943\pi$
$270$		0			0
$271$			173.665i			0.640831i	0.947277	+	0.320416i	$0.103823\pi$
$271$			173.665i			0.640831i	−0.947277	+	0.320416i	$0.896177\pi$
$272$		0			0
$273$	−4.82520			−0.0176747
$274$		0			0
$275$		0			0
$276$		0			0
$277$			92.4335i			0.333695i	0.985983	+	0.166848i	$0.0533588\pi$
$277$			92.4335i			0.333695i	−0.985983	+	0.166848i	$0.946641\pi$
$278$		0			0
$279$	−88.8746	+	179.636i	−0.318547	+	0.643857i
$280$		0			0
$281$	−86.6016			−0.308191			−0.154095	−	0.988056i	$-0.549246\pi$
$281$	−86.6016			−0.308191			−0.154095	+	0.988056i	$0.549246\pi$
$282$		0			0
$283$	−368.011			−1.30039			−0.650196	−	0.759767i	$-0.725313\pi$
$283$	−368.011			−1.30039			−0.650196	+	0.759767i	$0.725313\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−11.0845			−0.0386220
$288$		0			0
$289$	285.527			0.987982
$290$		0			0
$291$			256.909i			0.882848i
$292$		0			0
$293$	327.750			1.11860			0.559301	−	0.828965i	$-0.311070\pi$
$293$	327.750			1.11860			0.559301	+	0.828965i	$0.311070\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	49.5991			0.167000
$298$		0			0
$299$	50.9647			0.170451
$300$		0			0
$301$		−	76.2005i		−	0.253158i
$302$		0			0
$303$		−	202.091i		−	0.666967i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−61.5020			−0.200332			−0.100166	−	0.994971i	$-0.531937\pi$
$307$	−61.5020			−0.200332			−0.100166	+	0.994971i	$0.531937\pi$
$308$		0			0
$309$			56.1231i			0.181628i
$310$		0			0
$311$	379.929			1.22164			0.610819	−	0.791771i	$-0.290840\pi$
$311$	379.929			1.22164			0.610819	+	0.791771i	$0.290840\pi$
$312$		0			0
$313$		−	592.072i		−	1.89160i	−0.324744	−	0.945802i	$-0.605278\pi$
$313$		−	592.072i		−	1.89160i	0.324744	−	0.945802i	$-0.394722\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	120.174			0.379098			0.189549	−	0.981871i	$-0.439297\pi$
$317$	120.174			0.379098			0.189549	+	0.981871i	$0.439297\pi$
$318$		0			0
$319$	91.0651			0.285470
$320$		0			0
$321$		−	279.124i		−	0.869545i
$322$		0			0
$323$		−	12.3418i		−	0.0382099i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			25.3782i			0.0776091i
$328$		0			0
$329$	92.1179			0.279994
$330$		0			0
$331$		−	174.309i		−	0.526613i	−0.964712	−	0.263306i	$-0.915187\pi$
$331$		−	174.309i		−	0.526613i	0.964712	−	0.263306i	$-0.0848130\pi$
$332$		0			0
$333$			262.620i			0.788648i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			286.114i			0.849002i	0.905428	+	0.424501i	$0.139550\pi$
$337$			286.114i			0.849002i	−0.905428	+	0.424501i	$0.860450\pi$
$338$		0			0
$339$			89.4362i			0.263824i
$340$		0			0
$341$	55.9704	+	27.6913i	0.164136	+	0.0812060i
$342$		0			0
$343$	−153.285			−0.446895
$344$		0			0
$345$		0			0
$346$		0			0
$347$			439.597i			1.26685i	0.773804	+	0.633425i	$0.218352\pi$
$347$			439.597i			1.26685i	−0.773804	+	0.633425i	$0.781648\pi$
$348$		0			0
$349$	434.435			1.24480			0.622399	−	0.782700i	$-0.286158\pi$
$349$	434.435			1.24480			0.622399	+	0.782700i	$0.286158\pi$
$350$		0			0
$351$	−46.4521			−0.132342
$352$		0			0
$353$			39.0955i			0.110752i	0.998466	+	0.0553760i	$0.0176358\pi$
$353$			39.0955i			0.110752i	−0.998466	+	0.0553760i	$0.982364\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	4.76658			0.0133518
$358$		0			0
$359$	112.621			0.313708			0.156854	−	0.987622i	$-0.449865\pi$
$359$	112.621			0.313708			0.156854	+	0.987622i	$0.449865\pi$
$360$		0			0
$361$	−317.144			−0.878516
$362$		0			0
$363$			186.187i			0.512911i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			288.059i			0.784901i	0.919773	+	0.392451i	$0.128373\pi$
$367$			288.059i			0.784901i	−0.919773	+	0.392451i	$0.871627\pi$
$368$		0			0
$369$	−44.6099			−0.120894
$370$		0			0
$371$			0.0398860i			0.000107510i
$372$		0			0
$373$	−276.887			−0.742323			−0.371162	−	0.928568i	$-0.621040\pi$
$373$	−276.887			−0.742323			−0.371162	+	0.928568i	$0.621040\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−85.2871			−0.226226
$378$		0			0
$379$	220.436			0.581625			0.290812	−	0.956780i	$-0.406074\pi$
$379$	220.436			0.581625			0.290812	+	0.956780i	$0.406074\pi$
$380$		0			0
$381$	72.5916			0.190529
$382$		0			0
$383$		−	175.051i		−	0.457053i	−0.973538	−	0.228527i	$-0.926609\pi$
$383$		−	175.051i		−	0.457053i	0.973538	−	0.228527i	$-0.0733908\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	306.670i		−	0.792430i
$388$		0			0
$389$			316.942i			0.814762i	0.913258	+	0.407381i	$0.133558\pi$
$389$			316.942i			0.814762i	−0.913258	+	0.407381i	$0.866442\pi$
$390$		0			0
$391$	−50.3456			−0.128761
$392$		0			0
$393$		−	70.6762i		−	0.179838i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	251.079			0.632440			0.316220	−	0.948686i	$-0.397586\pi$
$397$	251.079			0.632440			0.316220	+	0.948686i	$0.397586\pi$
$398$		0			0
$399$			16.9377i			0.0424503i
$400$		0			0
$401$		−	534.043i		−	1.33178i	−0.746050	−	0.665890i	$-0.768052\pi$
$401$		−	534.043i		−	1.33178i	0.746050	−	0.665890i	$-0.231948\pi$
$402$		0			0
$403$	−52.4192	−	25.9343i	−0.130072	−	0.0643531i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	81.8262			0.201047
$408$		0			0
$409$			651.012i			1.59172i	0.605483	+	0.795859i	$0.292980\pi$
$409$			651.012i			1.59172i	−0.605483	+	0.795859i	$0.707020\pi$
$410$		0			0
$411$	−98.0333			−0.238524
$412$		0			0
$413$	118.201			0.286201
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−186.897			−0.448195
$418$		0			0
$419$	−9.97920			−0.0238167			−0.0119083	−	0.999929i	$-0.503791\pi$
$419$	−9.97920			−0.0238167			−0.0119083	+	0.999929i	$0.503791\pi$
$420$		0			0
$421$	99.2573			0.235765			0.117883	−	0.993028i	$-0.462389\pi$
$421$	99.2573			0.235765			0.117883	+	0.993028i	$0.462389\pi$
$422$		0			0
$423$	370.731			0.876432
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	131.103i		−	0.307032i
$428$		0			0
$429$			6.05054i			0.0141038i
$430$		0			0
$431$	−276.377			−0.641246			−0.320623	−	0.947207i	$-0.603892\pi$
$431$	−276.377			−0.641246			−0.320623	+	0.947207i	$0.603892\pi$
$432$		0			0
$433$		−	420.217i		−	0.970479i	−0.874381	−	0.485240i	$-0.838732\pi$
$433$		−	420.217i		−	0.970479i	0.874381	−	0.485240i	$-0.161268\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	178.899i		−	0.409380i
$438$		0			0
$439$	−261.937			−0.596669			−0.298334	−	0.954461i	$-0.596431\pi$
$439$	−261.937			−0.596669			−0.298334	+	0.954461i	$0.596431\pi$
$440$		0			0
$441$	−300.107			−0.680516
$442$		0			0
$443$	−673.290			−1.51984			−0.759921	−	0.650015i	$-0.774763\pi$
$443$	−673.290			−1.51984			−0.759921	+	0.650015i	$0.774763\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	87.3563i		−	0.195428i
$448$		0			0
$449$			358.542i			0.798534i	0.916835	+	0.399267i	$0.130735\pi$
$449$			358.542i			0.798534i	−0.916835	+	0.399267i	$0.869265\pi$
$450$		0			0
$451$			13.8994i			0.0308191i
$452$		0			0
$453$	−370.902			−0.818768
$454$		0			0
$455$		0			0
$456$		0			0
$457$			525.488i			1.14986i	0.818201	+	0.574932i	$0.194972\pi$
$457$			525.488i			1.14986i	−0.818201	+	0.574932i	$0.805028\pi$
$458$		0			0
$459$	45.8878			0.0999734
$460$		0			0
$461$			101.046i			0.219189i	0.993976	+	0.109594i	$0.0349552\pi$
$461$			101.046i			0.219189i	−0.993976	+	0.109594i	$0.965045\pi$
$462$		0			0
$463$		−	31.6061i		−	0.0682638i	−0.999417	−	0.0341319i	$-0.989133\pi$
$463$		−	31.6061i		−	0.0682638i	0.999417	−	0.0341319i	$-0.0108666\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	204.987			0.438944			0.219472	−	0.975619i	$-0.429567\pi$
$467$	204.987			0.438944			0.219472	+	0.975619i	$0.429567\pi$
$468$		0			0
$469$	80.4683			0.171574
$470$		0			0
$471$			22.5712i			0.0479218i
$472$		0			0
$473$	−95.5513			−0.202011
$474$		0			0
$475$		0			0
$476$		0			0
$477$			0.160522i			0.000336525i
$478$		0			0
$479$	44.6366			0.0931870			0.0465935	−	0.998914i	$-0.485163\pi$
$479$	44.6366			0.0931870			0.0465935	+	0.998914i	$0.485163\pi$
$480$		0			0
$481$	−76.6345			−0.159323
$482$		0			0
$483$	69.0934			0.143051
$484$		0			0
$485$		0			0
$486$		0			0
$487$			487.179i			1.00037i	0.865919	+	0.500184i	$0.166734\pi$
$487$			487.179i			1.00037i	−0.865919	+	0.500184i	$0.833266\pi$
$488$		0			0
$489$			144.117i			0.294719i
$490$		0			0
$491$		−	85.9961i		−	0.175145i	−0.996158	−	0.0875724i	$-0.972089\pi$
$491$		−	85.9961i		−	0.175145i	0.996158	−	0.0875724i	$-0.0279109\pi$
$492$		0			0
$493$	84.2511			0.170895
$494$		0			0
$495$		0			0
$496$		0			0
$497$	125.124			0.251758
$498$		0			0
$499$			500.280i			1.00256i	0.865284	+	0.501282i	$0.167138\pi$
$499$			500.280i			1.00256i	−0.865284	+	0.501282i	$0.832862\pi$
$500$		0			0
$501$	296.088			0.590994
$502$		0			0
$503$	−149.584			−0.297384			−0.148692	−	0.988884i	$-0.547506\pi$
$503$	−149.584			−0.297384			−0.148692	+	0.988884i	$0.547506\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			263.403i			0.519532i
$508$		0			0
$509$		−	575.053i		−	1.12977i	−0.825170	−	0.564885i	$-0.808921\pi$
$509$		−	575.053i		−	1.12977i	0.825170	−	0.564885i	$-0.191079\pi$
$510$		0			0
$511$		−	193.043i		−	0.377774i
$512$		0			0
$513$			163.059i			0.317853i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	115.511i		−	0.223425i
$518$		0			0
$519$			147.487i			0.284176i
$520$		0			0
$521$	−525.572			−1.00878			−0.504388	−	0.863477i	$-0.668282\pi$
$521$	−525.572			−1.00878			−0.504388	+	0.863477i	$0.668282\pi$
$522$		0			0
$523$		−	788.978i		−	1.50856i	−0.656551	−	0.754281i	$-0.727985\pi$
$523$		−	788.978i		−	1.50856i	0.656551	−	0.754281i	$-0.272015\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	51.7824	+	25.6192i	0.0982589	+	0.0486134i
$528$		0			0
$529$	−200.779			−0.379545
$530$		0			0
$531$	475.703			0.895862
$532$		0			0
$533$		−	13.0175i		−	0.0244231i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	7.80620			0.0145367
$538$		0			0
$539$			93.5065i			0.173481i
$540$		0			0
$541$	562.368			1.03950			0.519749	−	0.854319i	$-0.326026\pi$
$541$	562.368			1.03950			0.519749	+	0.854319i	$0.326026\pi$
$542$		0			0
$543$	−66.6394			−0.122724
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−389.572			−0.712197			−0.356098	−	0.934448i	$-0.615893\pi$
$547$	−389.572			−0.712197			−0.356098	+	0.934448i	$0.615893\pi$
$548$		0			0
$549$		−	527.626i		−	0.961067i
$550$		0			0
$551$			299.379i			0.543338i
$552$		0			0
$553$			170.407i			0.308150i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	452.326i		−	0.812075i	−0.913856	−	0.406038i	$-0.866910\pi$
$557$		−	452.326i		−	0.812075i	0.913856	−	0.406038i	$-0.133090\pi$
$558$		0			0
$559$	89.4888			0.160087
$560$		0			0
$561$		−	5.97704i		−	0.0106543i
$562$		0			0
$563$	613.424			1.08956			0.544781	−	0.838578i	$-0.316613\pi$
$563$	613.424			1.08956			0.544781	+	0.838578i	$0.316613\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	30.4969			0.0537864
$568$		0			0
$569$		−	606.320i		−	1.06559i	−0.846245	−	0.532794i	$-0.821142\pi$
$569$		−	606.320i		−	1.06559i	0.846245	−	0.532794i	$-0.178858\pi$
$570$		0			0
$571$			506.776i			0.887523i	0.896145	+	0.443762i	$0.146356\pi$
$571$			506.776i			0.887523i	−0.896145	+	0.443762i	$0.853644\pi$
$572$		0			0
$573$			361.354i			0.630636i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	921.880			1.59771			0.798856	−	0.601522i	$-0.205439\pi$
$577$	921.880			1.59771			0.798856	+	0.601522i	$0.205439\pi$
$578$		0			0
$579$			15.9140i			0.0274853i
$580$		0			0
$581$			170.154i			0.292863i
$582$		0			0
$583$	0.0500150			8.57890e−5
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	20.5955i		−	0.0350861i	−0.999846	−	0.0175430i	$-0.994416\pi$
$587$		−	20.5955i		−	0.0350861i	0.999846	−	0.0175430i	$-0.00558441\pi$
$588$		0			0
$589$	−91.0359	+	184.005i	−0.154560	+	0.312402i
$590$		0			0
$591$	429.298			0.726393
$592$		0			0
$593$	−27.7290			−0.0467606			−0.0233803	−	0.999727i	$-0.507443\pi$
$593$	−27.7290			−0.0467606			−0.0233803	+	0.999727i	$0.507443\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−328.358			−0.550014
$598$		0			0
$599$	−120.595			−0.201327			−0.100664	−	0.994921i	$-0.532097\pi$
$599$	−120.595			−0.201327			−0.100664	+	0.994921i	$0.532097\pi$
$600$		0			0
$601$		−	339.888i		−	0.565538i	−0.959188	−	0.282769i	$-0.908747\pi$
$601$		−	339.888i		−	0.565538i	0.959188	−	0.282769i	$-0.0912530\pi$
$602$		0			0
$603$	323.846			0.537059
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−473.516			−0.780092			−0.390046	−	0.920795i	$-0.627541\pi$
$607$	−473.516			−0.780092			−0.390046	+	0.920795i	$0.627541\pi$
$608$		0			0
$609$	−115.625			−0.189860
$610$		0			0
$611$			108.182i			0.177057i
$612$		0			0
$613$		−	452.885i		−	0.738801i	−0.929270	−	0.369400i	$-0.879563\pi$
$613$		−	452.885i		−	0.738801i	0.929270	−	0.369400i	$-0.120437\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−416.979			−0.675817			−0.337909	−	0.941179i	$-0.609719\pi$
$617$	−416.979			−0.675817			−0.337909	+	0.941179i	$0.609719\pi$
$618$		0			0
$619$			281.767i			0.455197i	0.973755	+	0.227599i	$0.0730874\pi$
$619$			281.767i			0.455197i	−0.973755	+	0.227599i	$0.926913\pi$
$620$		0			0
$621$	665.161			1.07111
$622$		0			0
$623$			123.349i			0.197992i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	21.2389			0.0338739
$628$		0			0
$629$	75.7035			0.120355
$630$		0			0
$631$			323.375i			0.512480i	0.966613	+	0.256240i	$0.0824837\pi$
$631$			323.375i			0.512480i	−0.966613	+	0.256240i	$0.917516\pi$
$632$		0			0
$633$			185.989i			0.293822i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	87.5736i		−	0.137478i
$638$		0			0
$639$	503.563			0.788048
$640$		0			0
$641$		−	22.6197i		−	0.0352882i	−0.999844	−	0.0176441i	$-0.994383\pi$
$641$		−	22.6197i		−	0.0352882i	0.999844	−	0.0176441i	$-0.00561658\pi$
$642$		0			0
$643$			461.777i			0.718161i	0.933307	+	0.359080i	$0.116910\pi$
$643$			461.777i			0.718161i	−0.933307	+	0.359080i	$0.883090\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	689.341i		−	1.06544i	−0.846291	−	0.532721i	$-0.821170\pi$
$647$		−	689.341i		−	1.06544i	0.846291	−	0.532721i	$-0.178830\pi$
$648$		0			0
$649$		−	148.218i		−	0.228379i
$650$		0			0
$651$	−71.0652	−	35.1594i	−0.109163	−	0.0540082i
$652$		0			0
$653$	572.698			0.877026			0.438513	−	0.898725i	$-0.355505\pi$
$653$	572.698			0.877026			0.438513	+	0.898725i	$0.355505\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	776.904i		−	1.18250i
$658$		0			0
$659$	−228.580			−0.346859			−0.173429	−	0.984846i	$-0.555485\pi$
$659$	−228.580			−0.346859			−0.173429	+	0.984846i	$0.555485\pi$
$660$		0			0
$661$	−483.184			−0.730990			−0.365495	−	0.930813i	$-0.619100\pi$
$661$	−483.184			−0.730990			−0.365495	+	0.930813i	$0.619100\pi$
$662$		0			0
$663$			5.59780i			0.00844314i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1221.25			1.83096
$668$		0			0
$669$	−562.059			−0.840148
$670$		0			0
$671$	−164.396			−0.245001
$672$		0			0
$673$		−	139.495i		−	0.207273i	−0.994615	−	0.103637i	$-0.966952\pi$
$673$		−	139.495i		−	0.207273i	0.994615	−	0.103637i	$-0.0330479\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	613.264i		−	0.905856i	−0.891547	−	0.452928i	$-0.850380\pi$
$677$		−	613.264i		−	0.905856i	0.891547	−	0.452928i	$-0.149620\pi$
$678$		0			0
$679$	259.218			0.381764
$680$		0			0
$681$		−	349.802i		−	0.513659i
$682$		0			0
$683$	530.112			0.776153			0.388076	−	0.921627i	$-0.373140\pi$
$683$	530.112			0.776153			0.388076	+	0.921627i	$0.373140\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−463.529			−0.674715
$688$		0			0
$689$	−0.0468416			−6.79849e−5
$690$		0			0
$691$	−992.386			−1.43616			−0.718079	−	0.695961i	$-0.754979\pi$
$691$	−992.386			−1.43616			−0.718079	+	0.695961i	$0.754979\pi$
$692$		0			0
$693$		−	20.9211i		−	0.0301891i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			12.8594i			0.0184496i
$698$		0			0
$699$		−	506.568i		−	0.724704i
$700$		0			0
$701$	−585.290			−0.834936			−0.417468	−	0.908692i	$-0.637082\pi$
$701$	−585.290			−0.834936			−0.417468	+	0.908692i	$0.637082\pi$
$702$		0			0
$703$			269.006i			0.382655i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−203.907			−0.288412
$708$		0			0
$709$		−	971.026i		−	1.36957i	−0.728745	−	0.684786i	$-0.759896\pi$
$709$		−	971.026i		−	1.36957i	0.728745	−	0.684786i	$-0.240104\pi$
$710$		0			0
$711$			685.807i			0.964567i
$712$		0			0
$713$	750.606	+	371.361i	1.05274	+	0.520842i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−169.128			−0.235883
$718$		0			0
$719$		−	566.411i		−	0.787776i	−0.919158	−	0.393888i	$-0.871130\pi$
$719$		−	566.411i		−	0.787776i	0.919158	−	0.393888i	$-0.128870\pi$
$720$		0			0
$721$	56.6275			0.0785402
$722$		0			0
$723$	414.490			0.573292
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−509.485			−0.700804			−0.350402	−	0.936599i	$-0.613955\pi$
$727$	−509.485			−0.700804			−0.350402	+	0.936599i	$0.613955\pi$
$728$		0			0
$729$	−230.083			−0.315614
$730$		0			0
$731$	−88.4017			−0.120933
$732$		0			0
$733$	771.841			1.05299			0.526494	−	0.850179i	$-0.323506\pi$
$733$	771.841			1.05299			0.526494	+	0.850179i	$0.323506\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	100.903i		−	0.136910i
$738$		0			0
$739$		−	276.135i		−	0.373661i	−0.982392	−	0.186830i	$-0.940178\pi$
$739$		−	276.135i		−	0.373661i	0.982392	−	0.186830i	$-0.0598215\pi$
$740$		0			0
$741$	−19.8913			−0.0268439
$742$		0			0
$743$			477.767i			0.643024i	0.946905	+	0.321512i	$0.104191\pi$
$743$			477.767i			0.643024i	−0.946905	+	0.321512i	$0.895809\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			684.787i			0.916716i
$748$		0			0
$749$	−281.633			−0.376012
$750$		0			0
$751$	443.995			0.591205			0.295602	−	0.955311i	$-0.404480\pi$
$751$	443.995			0.591205			0.295602	+	0.955311i	$0.404480\pi$
$752$		0			0
$753$	−766.389			−1.01778
$754$		0			0
$755$		0			0
$756$		0			0
$757$			686.038i			0.906259i	0.891445	+	0.453130i	$0.149693\pi$
$757$			686.038i			0.906259i	−0.891445	+	0.453130i	$0.850307\pi$
$758$		0			0
$759$		−	86.6394i		−	0.114149i
$760$		0			0
$761$		−	923.358i		−	1.21335i	−0.794951	−	0.606674i	$-0.792503\pi$
$761$		−	923.358i		−	1.21335i	0.794951	−	0.606674i	$-0.207497\pi$
$762$		0			0
$763$	25.6063			0.0335600
$764$		0			0
$765$		0			0
$766$		0			0
$767$			138.814i			0.180983i
$768$		0			0
$769$	−1113.70			−1.44824			−0.724120	−	0.689674i	$-0.757754\pi$
$769$	−1113.70			−1.44824			−0.724120	+	0.689674i	$0.757754\pi$
$770$		0			0
$771$			468.023i			0.607034i
$772$		0			0
$773$			1285.39i			1.66285i	0.555634	+	0.831427i	$0.312476\pi$
$773$			1285.39i			1.66285i	−0.555634	+	0.831427i	$0.687524\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−103.894			−0.133712
$778$		0			0
$779$	−45.6947			−0.0586582
$780$		0			0
$781$		−	156.898i		−	0.200894i
$782$		0			0
$783$	−1113.12			−1.42160
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	590.499i		−	0.750316i	−0.926961	−	0.375158i	$-0.877588\pi$
$787$		−	590.499i		−	0.750316i	0.926961	−	0.375158i	$-0.122412\pi$
$788$		0			0
$789$	30.3055			0.0384100
$790$		0			0
$791$	90.2401			0.114084
$792$		0			0
$793$	153.965			0.194155
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1170.71i		−	1.46890i	−0.678663	−	0.734450i	$-0.737440\pi$
$797$		−	1170.71i		−	1.46890i	0.678663	−	0.734450i	$-0.262560\pi$
$798$		0			0
$799$		−	106.868i		−	0.133752i
$800$		0			0
$801$			496.421i			0.619751i
$802$		0			0
$803$	−242.065			−0.301451
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−429.191			−0.531836
$808$		0			0
$809$		−	1543.31i		−	1.90767i	−0.300327	−	0.953836i	$-0.597096\pi$
$809$		−	1543.31i		−	1.90767i	0.300327	−	0.953836i	$-0.402904\pi$
$810$		0			0
$811$	1112.40			1.37164			0.685818	−	0.727773i	$-0.259445\pi$
$811$	1112.40			1.37164			0.685818	+	0.727773i	$0.259445\pi$
$812$		0			0
$813$	−276.497			−0.340095
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	314.128i		−	0.384490i
$818$		0			0
$819$			19.5936i			0.0239239i
$820$		0			0
$821$		−	417.348i		−	0.508341i	−0.967159	−	0.254170i	$-0.918198\pi$
$821$		−	417.348i		−	0.508341i	0.967159	−	0.254170i	$-0.0818024\pi$
$822$		0			0
$823$		−	683.988i		−	0.831091i	−0.909572	−	0.415546i	$-0.863591\pi$
$823$		−	683.988i		−	0.831091i	0.909572	−	0.415546i	$-0.136409\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			905.076i			1.09441i	0.836999	+	0.547204i	$0.184308\pi$
$827$			905.076i			1.09441i	−0.836999	+	0.547204i	$0.815692\pi$
$828$		0			0
$829$		−	1096.06i		−	1.32214i	−0.750323	−	0.661072i	$-0.770102\pi$
$829$		−	1096.06i		−	1.32214i	0.750323	−	0.661072i	$-0.229898\pi$
$830$		0			0
$831$	−147.166			−0.177095
$832$		0			0
$833$			86.5098i			0.103853i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−684.144	−	338.479i	−0.817376	−	0.404395i
$838$		0			0
$839$	701.813			0.836488			0.418244	−	0.908335i	$-0.362646\pi$
$839$	701.813			0.836488			0.418244	+	0.908335i	$0.362646\pi$
$840$		0			0
$841$	−1202.71			−1.43009
$842$		0			0
$843$		−	137.881i		−	0.163559i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	187.860			0.221795
$848$		0			0
$849$		−	585.919i		−	0.690129i
$850$		0			0
$851$	1097.35			1.28948
$852$		0			0
$853$	575.751			0.674972			0.337486	−	0.941331i	$-0.390423\pi$
$853$	575.751			0.674972			0.337486	+	0.941331i	$0.390423\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−950.070			−1.10860			−0.554300	−	0.832317i	$-0.687014\pi$
$857$	−950.070			−1.10860			−0.554300	+	0.832317i	$0.687014\pi$
$858$		0			0
$859$			15.9064i			0.0185174i	0.999957	+	0.00925870i	$0.00294718\pi$
$859$			15.9064i			0.0185174i	−0.999957	+	0.00925870i	$0.997053\pi$
$860$		0			0
$861$		−	17.6480i		−	0.0204970i
$862$		0			0
$863$		−	1171.72i		−	1.35772i	−0.734266	−	0.678862i	$-0.762474\pi$
$863$		−	1171.72i		−	1.35772i	0.734266	−	0.678862i	$-0.237526\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			454.595i			0.524331i
$868$		0			0
$869$	213.681			0.245894
$870$		0			0
$871$			94.5008i			0.108497i
$872$		0			0
$873$	1043.23			1.19499
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1211.65			1.38158			0.690792	−	0.723054i	$-0.257262\pi$
$877$	1211.65			1.38158			0.690792	+	0.723054i	$0.257262\pi$
$878$		0			0
$879$			521.820i			0.593652i
$880$		0			0
$881$		−	1517.57i		−	1.72255i	−0.508136	−	0.861277i	$-0.669665\pi$
$881$		−	1517.57i		−	1.72255i	0.508136	−	0.861277i	$-0.330335\pi$
$882$		0			0
$883$		−	323.245i		−	0.366076i	−0.983106	−	0.183038i	$-0.941407\pi$
$883$		−	323.245i		−	0.366076i	0.983106	−	0.183038i	$-0.0585932\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1452.08			1.63707			0.818536	−	0.574455i	$-0.194786\pi$
$887$	1452.08			1.63707			0.818536	+	0.574455i	$0.194786\pi$
$888$		0			0
$889$		−	73.2441i		−	0.0823893i
$890$		0			0
$891$		−	38.2415i		−	0.0429197i
$892$		0			0
$893$	379.746			0.425248
$894$		0			0
$895$		0			0
$896$		0			0
$897$			81.1423i			0.0904596i
$898$		0			0
$899$	−1256.10	−	621.455i	−1.39722	−	0.691273i
$900$		0			0
$901$	0.0462726			5.13569e−5
$902$		0			0
$903$	121.321			0.134353
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1110.78			1.22467			0.612337	−	0.790597i	$-0.290230\pi$
$907$	1110.78			1.22467			0.612337	+	0.790597i	$0.290230\pi$
$908$		0			0
$909$	−820.630			−0.902783
$910$		0			0
$911$		−	1576.75i		−	1.73079i	−0.501093	−	0.865393i	$-0.667069\pi$
$911$		−	1576.75i		−	1.73079i	0.501093	−	0.865393i	$-0.332931\pi$
$912$		0			0
$913$	213.364			0.233695
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−71.3115			−0.0777661
$918$		0			0
$919$	−1556.21			−1.69338			−0.846689	−	0.532088i	$-0.821408\pi$
$919$	−1556.21			−1.69338			−0.846689	+	0.532088i	$0.821408\pi$
$920$		0			0
$921$		−	97.9188i		−	0.106318i
$922$		0			0
$923$			146.943i			0.159202i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	227.899			0.245845
$928$		0			0
$929$		−	740.277i		−	0.796853i	−0.917200	−	0.398427i	$-0.869556\pi$
$929$		−	740.277i		−	0.796853i	0.917200	−	0.398427i	$-0.130444\pi$
$930$		0			0
$931$	−307.406			−0.330189
$932$		0			0
$933$			604.895i			0.648333i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1038.48			1.10830			0.554149	−	0.832417i	$-0.313044\pi$
$937$	1038.48			1.10830			0.554149	+	0.832417i	$0.313044\pi$
$938$		0			0
$939$	942.653			1.00389
$940$		0			0
$941$			146.282i			0.155454i	0.996975	+	0.0777271i	$0.0247663\pi$
$941$			146.282i			0.155454i	−0.996975	+	0.0777271i	$0.975234\pi$
$942$		0			0
$943$			186.401i			0.197669i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	185.321i		−	0.195692i	−0.995202	−	0.0978462i	$-0.968805\pi$
$947$		−	185.321i		−	0.195692i	0.995202	−	0.0978462i	$-0.0311953\pi$
$948$		0			0
$949$	226.706			0.238890
$950$		0			0
$951$			191.332i			0.201190i
$952$		0			0
$953$		−	655.282i		−	0.687599i	−0.939043	−	0.343799i	$-0.888286\pi$
$953$		−	655.282i		−	0.687599i	0.939043	−	0.343799i	$-0.111714\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			144.987i			0.151502i
$958$		0			0
$959$			98.9144i			0.103143i
$960$		0			0
$961$	−583.053	−	763.917i	−0.606715	−	0.794919i
$962$		0			0
$963$	−1133.44			−1.17698
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	306.500i		−	0.316960i	−0.987362	−	0.158480i	$-0.949341\pi$
$967$		−	306.500i		−	0.316960i	0.987362	−	0.158480i	$-0.0506593\pi$
$968$		0			0
$969$	19.6497			0.0202783
$970$		0			0
$971$	−521.154			−0.536719			−0.268359	−	0.963319i	$-0.586481\pi$
$971$	−521.154			−0.536719			−0.268359	+	0.963319i	$0.586481\pi$
$972$		0			0
$973$			188.577i			0.193810i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−957.638			−0.980183			−0.490091	−	0.871671i	$-0.663037\pi$
$977$	−957.638			−0.980183			−0.490091	+	0.871671i	$0.663037\pi$
$978$		0			0
$979$	154.673			0.157991
$980$		0			0
$981$	103.053			0.105049
$982$		0			0
$983$		−	1000.24i		−	1.01754i	−0.860902	−	0.508771i	$-0.830100\pi$
$983$		−	1000.24i		−	1.01754i	0.860902	−	0.508771i	$-0.169900\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			146.663i			0.148595i
$988$		0			0
$989$	−1281.42			−1.29567
$990$		0			0
$991$			755.163i			0.762021i	0.924571	+	0.381011i	$0.124424\pi$
$991$			755.163i			0.762021i	−0.924571	+	0.381011i	$0.875576\pi$
$992$		0			0
$993$	277.521			0.279478
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1913.31			−1.91907			−0.959533	−	0.281598i	$-0.909136\pi$
$997$	−1913.31			−1.91907			−0.959533	+	0.281598i	$0.909136\pi$
$998$		0			0
$999$	−1000.19			−1.00119

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3100.3.d.h.1301.12		24
5.2	odd	4		620.3.f.c.309.16	yes	24
5.3	odd	4		620.3.f.c.309.9	✓	24
5.4	even	2	inner	3100.3.d.h.1301.13		24
31.30	odd	2	inner	3100.3.d.h.1301.11		24
155.92	even	4		620.3.f.c.309.10	yes	24
155.123	even	4		620.3.f.c.309.15	yes	24
155.154	odd	2	inner	3100.3.d.h.1301.14		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
620.3.f.c.309.9	✓	24	5.3	odd	4
620.3.f.c.309.10	yes	24	155.92	even	4
620.3.f.c.309.15	yes	24	155.123	even	4
620.3.f.c.309.16	yes	24	5.2	odd	4
3100.3.d.h.1301.11		24	31.30	odd	2	inner
3100.3.d.h.1301.12		24	1.1	even	1	trivial
3100.3.d.h.1301.13		24	5.4	even	2	inner
3100.3.d.h.1301.14		24	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.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.59213i q^{3} +1.60644 q^{7} +6.46514 q^{9} +O(q^{10})\)	\(q+1.59213i q^{3} +1.60644 q^{7} +6.46514 q^{9} -2.01438i q^{11} +1.88658i q^{13} -1.86366i q^{17} +6.62236 q^{19} +2.55765i q^{21} -27.0144i q^{23} +24.6224i q^{27} +45.2074i q^{29} +(-13.7468 + 27.7854i) q^{31} +3.20715 q^{33} +40.6209i q^{37} -3.00367 q^{39} -6.90007 q^{41} -47.4345i q^{43} +57.3430 q^{47} -46.4194 q^{49} +2.96718 q^{51} +0.0248289i q^{53} +10.5436i q^{57} +73.5797 q^{59} -81.6110i q^{61} +10.3858 q^{63} +50.0912 q^{67} +43.0104 q^{69} +77.8890 q^{71} -120.168i q^{73} -3.23598i q^{77} +106.078i q^{79} +18.9842 q^{81} +105.920i q^{83} -71.9758 q^{87} +76.7843i q^{89} +3.03066i q^{91} +(-44.2378 - 21.8866i) q^{93} +161.362 q^{97} -13.0233i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 44 q^{9}+O(q^{10}) \) 24 * q - 44 * q^9	\( 24 q - 44 q^{9} - 8 q^{19} - 24 q^{31} + 280 q^{39} - 248 q^{41} + 644 q^{49} - 100 q^{51} - 152 q^{59} + 288 q^{69} - 352 q^{71} - 368 q^{81}+O(q^{100}) \) 24 * q - 44 * q^9 - 8 * q^19 - 24 * q^31 + 280 * q^39 - 248 * q^41 + 644 * q^49 - 100 * q^51 - 152 * q^59 + 288 * q^69 - 352 * q^71 - 368 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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