LMFDB - Embedded newform 4100.2.d.c.1149.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4100,2,Mod(1149,4100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4100, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4100.1149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4100 = 2^{2} \cdot 5^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4100.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.7386648287$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 4x^{5} + 14x^{4} - 14x^{3} + 8x^{2} + 4x + 1 $ x^8 - 2x^7 + 2x^6 + 4x^5 + 14x^4 - 14x^3 + 8x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 164)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1149.8
Root			$1.76639 - 1.76639i$ of defining polynomial
Character	$\chi$	$=$	4100.1149
Dual form			4100.2.d.c.1149.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.24028i q^{3} -0.858626i q^{7} -7.49944 q^{9} +O(q^{10})$	$q+3.24028i q^{3} -0.858626i q^{7} -7.49944 q^{9} +6.20694 q^{11} -1.41500i q^{13} -3.93332i q^{17} -3.82529 q^{19} +2.78219 q^{21} -3.06557i q^{23} -14.5795i q^{27} +8.48057 q^{29} +1.13225 q^{31} +20.1123i q^{33} +8.49944i q^{37} +4.58500 q^{39} -1.00000 q^{41} -5.34832i q^{43} -6.65528i q^{47} +6.26276 q^{49} +12.7451 q^{51} -6.41389i q^{53} -12.3950i q^{57} +3.06557 q^{59} +7.41500 q^{61} +6.43922i q^{63} +1.79306i q^{67} +9.93332 q^{69} -3.02422 q^{71} -0.632807i q^{73} -5.32944i q^{77} +14.3059 q^{79} +24.7433 q^{81} +8.76332i q^{83} +27.4795i q^{87} +7.89557 q^{89} -1.21496 q^{91} +3.66882i q^{93} +9.86664i q^{97} -46.5486 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9}+O(q^{10}) $ 8 * q - 24 * q^9	$ 8 q - 24 q^{9} + 8 q^{11} - 12 q^{19} + 8 q^{29} - 16 q^{31} + 48 q^{39} - 8 q^{41} - 32 q^{49} - 8 q^{51} - 24 q^{59} + 48 q^{61} + 56 q^{69} - 4 q^{71} + 36 q^{79} + 56 q^{81} - 8 q^{89} + 72 q^{91} - 116 q^{99}+O(q^{100}) $ 8 * q - 24 * q^9 + 8 * q^11 - 12 * q^19 + 8 * q^29 - 16 * q^31 + 48 * q^39 - 8 * q^41 - 32 * q^49 - 8 * q^51 - 24 * q^59 + 48 * q^61 + 56 * q^69 - 4 * q^71 + 36 * q^79 + 56 * q^81 - 8 * q^89 + 72 * q^91 - 116 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4100\mathbb{Z}\right)^\times$.

$n$	$1477$	$2051$	$3901$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	3.24028i	1.87078i	0.353619	+	0.935390i	$0.384951\pi$
$3$	3.24028i	1.87078i	−0.353619	+	0.935390i	$0.615049\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.858626i	− 0.324530i	−0.986747	−	0.162265i	$-0.948120\pi$
$7$	− 0.858626i	− 0.324530i	0.986747	−	0.162265i	$-0.0518800\pi$
$8$	0	0
$9$	−7.49944	−2.49981
$10$	0	0
$11$	6.20694	1.87146	0.935732	−	0.352712i	$-0.114740\pi$
$11$	6.20694	1.87146	0.935732	+	0.352712i	$0.114740\pi$
$12$	0	0
$13$	− 1.41500i	− 0.392450i	−0.980559	−	0.196225i	$-0.937132\pi$
$13$	− 1.41500i	− 0.392450i	0.980559	−	0.196225i	$-0.0628683\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.93332i	− 0.953970i	−0.878911	−	0.476985i	$-0.841730\pi$
$17$	− 3.93332i	− 0.953970i	0.878911	−	0.476985i	$-0.158270\pi$
$18$	0	0
$19$	−3.82529	−0.877581	−0.438790	−	0.898589i	$-0.644593\pi$
$19$	−3.82529	−0.877581	−0.438790	+	0.898589i	$0.644593\pi$
$20$	0	0
$21$	2.78219	0.607124
$22$	0	0
$23$	− 3.06557i	− 0.639215i	−0.947550	−	0.319608i	$-0.896449\pi$
$23$	− 3.06557i	− 0.639215i	0.947550	−	0.319608i	$-0.103551\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 14.5795i	− 2.80582i
$28$	0	0
$29$	8.48057	1.57480	0.787401	−	0.616441i	$-0.211426\pi$
$29$	8.48057	1.57480	0.787401	+	0.616441i	$0.211426\pi$
$30$	0	0
$31$	1.13225	0.203358	0.101679	−	0.994817i	$-0.467578\pi$
$31$	1.13225	0.203358	0.101679	+	0.994817i	$0.467578\pi$
$32$	0	0
$33$	20.1123i	3.50110i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.49944i	1.39730i	0.715464	+	0.698650i	$0.246216\pi$
$37$	8.49944i	1.39730i	−0.715464	+	0.698650i	$0.753784\pi$
$38$	0	0
$39$	4.58500	0.734188
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	− 5.34832i	− 0.815611i	−0.913069	−	0.407805i	$-0.866294\pi$
$43$	− 5.34832i	− 0.815611i	0.913069	−	0.407805i	$-0.133706\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6.65528i	− 0.970773i	−0.874300	−	0.485386i	$-0.838679\pi$
$47$	− 6.65528i	− 0.970773i	0.874300	−	0.485386i	$-0.161321\pi$
$48$	0	0
$49$	6.26276	0.894680
$50$	0	0
$51$	12.7451	1.78467
$52$	0	0
$53$	− 6.41389i	− 0.881015i	−0.897749	−	0.440508i	$-0.854798\pi$
$53$	− 6.41389i	− 0.881015i	0.897749	−	0.440508i	$-0.145202\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 12.3950i	− 1.64176i
$58$	0	0
$59$	3.06557	0.399103	0.199552	−	0.979887i	$-0.436051\pi$
$59$	3.06557	0.399103	0.199552	+	0.979887i	$0.436051\pi$
$60$	0	0
$61$	7.41500	0.949393	0.474697	−	0.880149i	$-0.342558\pi$
$61$	7.41500	0.949393	0.474697	+	0.880149i	$0.342558\pi$
$62$	0	0
$63$	6.43922i	0.811265i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.79306i	0.219057i	0.993984	+	0.109528i	$0.0349340\pi$
$67$	1.79306i	0.219057i	−0.993984	+	0.109528i	$0.965066\pi$
$68$	0	0
$69$	9.93332	1.19583
$70$	0	0
$71$	−3.02422	−0.358909	−0.179454	−	0.983766i	$-0.557433\pi$
$71$	−3.02422	−0.358909	−0.179454	+	0.983766i	$0.557433\pi$
$72$	0	0
$73$	− 0.632807i	− 0.0740645i	−0.999314	−	0.0370322i	$-0.988210\pi$
$73$	− 0.632807i	− 0.0740645i	0.999314	−	0.0370322i	$-0.0117904\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.32944i	− 0.607346i
$78$	0	0
$79$	14.3059	1.60953	0.804767	−	0.593591i	$-0.202290\pi$
$79$	14.3059	1.60953	0.804767	+	0.593591i	$0.202290\pi$
$80$	0	0
$81$	24.7433	2.74926
$82$	0	0
$83$	8.76332i	0.961899i	0.876748	+	0.480950i	$0.159708\pi$
$83$	8.76332i	0.961899i	−0.876748	+	0.480950i	$0.840292\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	27.4795i	2.94611i
$88$	0	0
$89$	7.89557	0.836929	0.418464	−	0.908233i	$-0.362569\pi$
$89$	7.89557	0.836929	0.418464	+	0.908233i	$0.362569\pi$
$90$	0	0
$91$	−1.21496	−0.127362
$92$	0	0
$93$	3.66882i	0.380439i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.86664i	1.00181i	0.865504	+	0.500903i	$0.166999\pi$
$97$	9.86664i	1.00181i	−0.865504	+	0.500903i	$0.833001\pi$
$98$	0	0
$99$	−46.5486	−4.67831
$100$	0	0
$101$	−4.51832	−0.449590	−0.224795	−	0.974406i	$-0.572171\pi$
$101$	−4.51832	−0.449590	−0.224795	+	0.974406i	$0.572171\pi$
$102$	0	0
$103$	16.4139i	1.61731i	0.588284	+	0.808654i	$0.299804\pi$
$103$	16.4139i	1.61731i	−0.588284	+	0.808654i	$0.700196\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 16.0645i	− 1.55301i	−0.630111	−	0.776505i	$-0.716991\pi$
$107$	− 16.0645i	− 1.55301i	0.630111	−	0.776505i	$-0.283009\pi$
$108$	0	0
$109$	5.61282	0.537611	0.268805	−	0.963195i	$-0.413371\pi$
$109$	5.61282	0.537611	0.268805	+	0.963195i	$0.413371\pi$
$110$	0	0
$111$	−27.5406	−2.61404
$112$	0	0
$113$	− 5.63170i	− 0.529785i	−0.964278	−	0.264893i	$-0.914663\pi$
$113$	− 5.63170i	− 0.529785i	0.964278	−	0.264893i	$-0.0853365\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	10.6117i	0.981053i
$118$	0	0
$119$	−3.37725	−0.309592
$120$	0	0
$121$	27.5262	2.50238
$122$	0	0
$123$	− 3.24028i	− 0.292167i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.69775i	0.150651i	0.997159	+	0.0753254i	$0.0239995\pi$
$127$	1.69775i	0.150651i	−0.997159	+	0.0753254i	$0.976000\pi$
$128$	0	0
$129$	17.3301	1.52583
$130$	0	0
$131$	−2.48057	−0.216728	−0.108364	−	0.994111i	$-0.534561\pi$
$131$	−2.48057	−0.216728	−0.108364	+	0.994111i	$0.534561\pi$
$132$	0	0
$133$	3.28449i	0.284801i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.76332i	0.236086i	0.993008	+	0.118043i	$0.0376621\pi$
$137$	2.76332i	0.236086i	−0.993008	+	0.118043i	$0.962338\pi$
$138$	0	0
$139$	−8.99889	−0.763276	−0.381638	−	0.924312i	$-0.624640\pi$
$139$	−8.99889	−0.763276	−0.381638	+	0.924312i	$0.624640\pi$
$140$	0	0
$141$	21.5650	1.81610
$142$	0	0
$143$	− 8.78282i	− 0.734456i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	20.2931i	1.67375i
$148$	0	0
$149$	8.48057	0.694755	0.347378	−	0.937725i	$-0.387072\pi$
$149$	8.48057	0.694755	0.347378	+	0.937725i	$0.387072\pi$
$150$	0	0
$151$	−7.58971	−0.617642	−0.308821	−	0.951120i	$-0.599934\pi$
$151$	−7.58971	−0.617642	−0.308821	+	0.951120i	$0.599934\pi$
$152$	0	0
$153$	29.4977i	2.38475i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 6.78282i	− 0.541328i	−0.962674	−	0.270664i	$-0.912757\pi$
$157$	− 6.78282i	− 0.541328i	0.962674	−	0.270664i	$-0.0872433\pi$
$158$	0	0
$159$	20.7828	1.64818
$160$	0	0
$161$	−2.63218	−0.207445
$162$	0	0
$163$	− 15.0278i	− 1.17707i	−0.808472	−	0.588535i	$-0.799705\pi$
$163$	− 15.0278i	− 1.17707i	0.808472	−	0.588535i	$-0.200295\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.79306i	0.448280i	0.974557	+	0.224140i	$0.0719573\pi$
$167$	5.79306i	0.448280i	−0.974557	+	0.224140i	$0.928043\pi$
$168$	0	0
$169$	10.9978	0.845983
$170$	0	0
$171$	28.6875	2.19379
$172$	0	0
$173$	5.43450i	0.413178i	0.978428	+	0.206589i	$0.0662362\pi$
$173$	5.43450i	0.413178i	−0.978428	+	0.206589i	$0.933764\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	9.93332i	0.746634i
$178$	0	0
$179$	−8.10251	−0.605610	−0.302805	−	0.953052i	$-0.597923\pi$
$179$	−8.10251	−0.605610	−0.302805	+	0.953052i	$0.597923\pi$
$180$	0	0
$181$	−9.24389	−0.687093	−0.343546	−	0.939136i	$-0.611628\pi$
$181$	−9.24389	−0.687093	−0.343546	+	0.939136i	$0.611628\pi$
$182$	0	0
$183$	24.0267i	1.77611i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 24.4139i	− 1.78532i
$188$	0	0
$189$	−12.5183	−0.910574
$190$	0	0
$191$	24.3381	1.76104	0.880521	−	0.474007i	$-0.157193\pi$
$191$	24.3381	1.76104	0.880521	+	0.474007i	$0.157193\pi$
$192$	0	0
$193$	− 13.2634i	− 0.954720i	−0.878708	−	0.477360i	$-0.841594\pi$
$193$	− 13.2634i	− 0.954720i	0.878708	−	0.477360i	$-0.158406\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 18.1116i	− 1.29040i	−0.764013	−	0.645200i	$-0.776774\pi$
$197$	− 18.1116i	− 1.29040i	0.764013	−	0.645200i	$-0.223226\pi$
$198$	0	0
$199$	−19.7726	−1.40164	−0.700821	−	0.713337i	$-0.747183\pi$
$199$	−19.7726	−1.40164	−0.700821	+	0.713337i	$0.747183\pi$
$200$	0	0
$201$	−5.81001	−0.409807
$202$	0	0
$203$	− 7.28164i	− 0.511071i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	22.9901i	1.59792i
$208$	0	0
$209$	−23.7433	−1.64236
$210$	0	0
$211$	1.53924	0.105966	0.0529828	−	0.998595i	$-0.483127\pi$
$211$	1.53924	0.105966	0.0529828	+	0.998595i	$0.483127\pi$
$212$	0	0
$213$	− 9.79933i	− 0.671439i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 0.972180i	− 0.0659959i
$218$	0	0
$219$	2.05047	0.138558
$220$	0	0
$221$	−5.56564	−0.374386
$222$	0	0
$223$	27.6578i	1.85210i	0.377399	+	0.926051i	$0.376819\pi$
$223$	27.6578i	1.85210i	−0.377399	+	0.926051i	$0.623181\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.79635i	− 0.119228i	−0.998221	−	0.0596141i	$-0.981013\pi$
$227$	− 1.79635i	− 0.119228i	0.998221	−	0.0596141i	$-0.0189870\pi$
$228$	0	0
$229$	−21.6128	−1.42822	−0.714108	−	0.700036i	$-0.753167\pi$
$229$	−21.6128	−1.42822	−0.714108	+	0.700036i	$0.753167\pi$
$230$	0	0
$231$	17.2689	1.13621
$232$	0	0
$233$	18.3312i	1.20092i	0.799656	+	0.600458i	$0.205015\pi$
$233$	18.3312i	1.20092i	−0.799656	+	0.600458i	$0.794985\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	46.3550i	3.01108i
$238$	0	0
$239$	−11.9586	−0.773541	−0.386770	−	0.922176i	$-0.626409\pi$
$239$	−11.9586	−0.773541	−0.386770	+	0.922176i	$0.626409\pi$
$240$	0	0
$241$	7.41500	0.477642	0.238821	−	0.971064i	$-0.423239\pi$
$241$	7.41500	0.477642	0.238821	+	0.971064i	$0.423239\pi$
$242$	0	0
$243$	36.4370i	2.33743i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.41278i	0.344407i
$248$	0	0
$249$	−28.3956	−1.79950
$250$	0	0
$251$	−1.91507	−0.120878	−0.0604392	−	0.998172i	$-0.519250\pi$
$251$	−1.91507	−0.120878	−0.0604392	+	0.998172i	$0.519250\pi$
$252$	0	0
$253$	− 19.0278i	− 1.19627i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.93443i	0.183045i	0.995803	+	0.0915224i	$0.0291733\pi$
$257$	2.93443i	0.183045i	−0.995803	+	0.0915224i	$0.970827\pi$
$258$	0	0
$259$	7.29784	0.453466
$260$	0	0
$261$	−63.5996	−3.93671
$262$	0	0
$263$	3.10692i	0.191581i	0.995402	+	0.0957905i	$0.0305379\pi$
$263$	3.10692i	0.191581i	−0.995402	+	0.0957905i	$0.969462\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	25.5839i	1.56571i
$268$	0	0
$269$	10.2450	0.624649	0.312324	−	0.949976i	$-0.398892\pi$
$269$	10.2450	0.624649	0.312324	+	0.949976i	$0.398892\pi$
$270$	0	0
$271$	−11.1989	−0.680287	−0.340143	−	0.940374i	$-0.610476\pi$
$271$	−11.1989	−0.680287	−0.340143	+	0.940374i	$0.610476\pi$
$272$	0	0
$273$	− 3.93680i	− 0.238266i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.52838i	0.151915i	0.997111	+	0.0759577i	$0.0242014\pi$
$277$	2.52838i	0.151915i	−0.997111	+	0.0759577i	$0.975799\pi$
$278$	0	0
$279$	−8.49125	−0.508358
$280$	0	0
$281$	−23.5656	−1.40581	−0.702904	−	0.711285i	$-0.748114\pi$
$281$	−23.5656	−1.40581	−0.702904	+	0.711285i	$0.748114\pi$
$282$	0	0
$283$	− 8.99889i	− 0.534928i	−0.963568	−	0.267464i	$-0.913814\pi$
$283$	− 8.99889i	− 0.534928i	0.963568	−	0.267464i	$-0.0861857\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.858626i	0.0506831i
$288$	0	0
$289$	1.52900	0.0899415
$290$	0	0
$291$	−31.9707	−1.87416
$292$	0	0
$293$	− 3.17721i	− 0.185614i	−0.995684	−	0.0928072i	$-0.970416\pi$
$293$	− 3.17721i	− 0.185614i	0.995684	−	0.0928072i	$-0.0295840\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 90.4940i	− 5.25100i
$298$	0	0
$299$	−4.33778	−0.250860
$300$	0	0
$301$	−4.59220	−0.264690
$302$	0	0
$303$	− 14.6406i	− 0.841083i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 17.5839i	− 1.00357i	−0.864994	−	0.501783i	$-0.832678\pi$
$307$	− 17.5839i	− 1.00357i	0.864994	−	0.501783i	$-0.167322\pi$
$308$	0	0
$309$	−53.1857	−3.02563
$310$	0	0
$311$	1.55749	0.0883169	0.0441584	−	0.999025i	$-0.485939\pi$
$311$	1.55749	0.0883169	0.0441584	+	0.999025i	$0.485939\pi$
$312$	0	0
$313$	− 20.5255i	− 1.16017i	−0.814556	−	0.580086i	$-0.803019\pi$
$313$	− 20.5255i	− 1.16017i	0.814556	−	0.580086i	$-0.196981\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.1094i	1.46645i	0.679986	+	0.733225i	$0.261986\pi$
$317$	26.1094i	1.46645i	−0.679986	+	0.733225i	$0.738014\pi$
$318$	0	0
$319$	52.6384	2.94719
$320$	0	0
$321$	52.0534	2.90534
$322$	0	0
$323$	15.0461i	0.837185i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	18.1871i	1.00575i
$328$	0	0
$329$	−5.71440	−0.315045
$330$	0	0
$331$	3.68862	0.202745	0.101373	−	0.994849i	$-0.467677\pi$
$331$	3.68862	0.202745	0.101373	+	0.994849i	$0.467677\pi$
$332$	0	0
$333$	− 63.7411i	− 3.49299i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	30.9240i	1.68454i	0.539057	+	0.842269i	$0.318781\pi$
$337$	30.9240i	1.68454i	−0.539057	+	0.842269i	$0.681219\pi$
$338$	0	0
$339$	18.2483	0.991111
$340$	0	0
$341$	7.02782	0.380578
$342$	0	0
$343$	− 11.3878i	− 0.614881i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	27.1264i	1.45622i	0.685459	+	0.728111i	$0.259602\pi$
$347$	27.1264i	1.45622i	−0.685459	+	0.728111i	$0.740398\pi$
$348$	0	0
$349$	9.30051	0.497845	0.248922	−	0.968523i	$-0.419924\pi$
$349$	9.30051	0.497845	0.248922	+	0.968523i	$0.419924\pi$
$350$	0	0
$351$	−20.6300	−1.10115
$352$	0	0
$353$	− 13.3961i	− 0.713004i	−0.934295	−	0.356502i	$-0.883969\pi$
$353$	− 13.3961i	− 0.713004i	0.934295	−	0.356502i	$-0.116031\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 10.9432i	− 0.579178i
$358$	0	0
$359$	28.9611	1.52851	0.764255	−	0.644914i	$-0.223107\pi$
$359$	28.9611	1.52851	0.764255	+	0.644914i	$0.223107\pi$
$360$	0	0
$361$	−4.36719	−0.229852
$362$	0	0
$363$	89.1926i	4.68140i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 31.4795i	− 1.64321i	−0.570054	−	0.821607i	$-0.693078\pi$
$367$	− 31.4795i	− 1.64321i	0.570054	−	0.821607i	$-0.306922\pi$
$368$	0	0
$369$	7.49944	0.390405
$370$	0	0
$371$	−5.50713	−0.285916
$372$	0	0
$373$	15.5266i	0.803939i	0.915653	+	0.401969i	$0.131674\pi$
$373$	15.5266i	0.803939i	−0.915653	+	0.401969i	$0.868326\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 12.0000i	− 0.618031i
$378$	0	0
$379$	−1.97107	−0.101247	−0.0506235	−	0.998718i	$-0.516121\pi$
$379$	−1.97107	−0.101247	−0.0506235	+	0.998718i	$0.516121\pi$
$380$	0	0
$381$	−5.50119	−0.281834
$382$	0	0
$383$	12.8404i	0.656113i	0.944658	+	0.328056i	$0.106394\pi$
$383$	12.8404i	0.656113i	−0.944658	+	0.328056i	$0.893606\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	40.1094i	2.03888i
$388$	0	0
$389$	−1.03886	−0.0526724	−0.0263362	−	0.999653i	$-0.508384\pi$
$389$	−1.03886	−0.0526724	−0.0263362	+	0.999653i	$0.508384\pi$
$390$	0	0
$391$	−12.0579	−0.609792
$392$	0	0
$393$	− 8.03775i	− 0.405451i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 24.6406i	− 1.23668i	−0.785911	−	0.618339i	$-0.787806\pi$
$397$	− 24.6406i	− 1.23668i	0.785911	−	0.618339i	$-0.212194\pi$
$398$	0	0
$399$	−10.6427	−0.532800
$400$	0	0
$401$	0.397236	0.0198370	0.00991851	−	0.999951i	$-0.496843\pi$
$401$	0.397236	0.0198370	0.00991851	+	0.999951i	$0.496843\pi$
$402$	0	0
$403$	− 1.60213i	− 0.0798080i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	52.7556i	2.61500i
$408$	0	0
$409$	2.76395	0.136668	0.0683342	−	0.997662i	$-0.478232\pi$
$409$	2.76395	0.136668	0.0683342	+	0.997662i	$0.478232\pi$
$410$	0	0
$411$	−8.95393	−0.441665
$412$	0	0
$413$	− 2.63218i	− 0.129521i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 29.1590i	− 1.42792i
$418$	0	0
$419$	−1.48168	−0.0723848	−0.0361924	−	0.999345i	$-0.511523\pi$
$419$	−1.48168	−0.0723848	−0.0361924	+	0.999345i	$0.511523\pi$
$420$	0	0
$421$	−18.9039	−0.921319	−0.460659	−	0.887577i	$-0.652387\pi$
$421$	−18.9039	−0.921319	−0.460659	+	0.887577i	$0.652387\pi$
$422$	0	0
$423$	49.9109i	2.42675i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 6.36671i	− 0.308107i
$428$	0	0
$429$	28.4588	1.37401
$430$	0	0
$431$	−7.28164	−0.350744	−0.175372	−	0.984502i	$-0.556113\pi$
$431$	−7.28164	−0.350744	−0.175372	+	0.984502i	$0.556113\pi$
$432$	0	0
$433$	0.397865i	0.0191202i	0.999954	+	0.00956009i	$0.00304312\pi$
$433$	0.397865i	0.0191202i	−0.999954	+	0.00956009i	$0.996957\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	11.7267i	0.560963i
$438$	0	0
$439$	−33.9798	−1.62177	−0.810885	−	0.585206i	$-0.801014\pi$
$439$	−33.9798	−1.62177	−0.810885	+	0.585206i	$0.801014\pi$
$440$	0	0
$441$	−46.9672	−2.23653
$442$	0	0
$443$	− 16.2050i	− 0.769924i	−0.922932	−	0.384962i	$-0.874215\pi$
$443$	− 16.2050i	− 0.769924i	0.922932	−	0.384962i	$-0.125785\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	27.4795i	1.29973i
$448$	0	0
$449$	23.1118	1.09071	0.545356	−	0.838204i	$-0.316394\pi$
$449$	23.1118	1.09071	0.545356	+	0.838204i	$0.316394\pi$
$450$	0	0
$451$	−6.20694	−0.292274
$452$	0	0
$453$	− 24.5928i	− 1.15547i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	35.6301i	1.66671i	0.552741	+	0.833353i	$0.313582\pi$
$457$	35.6301i	1.66671i	−0.552741	+	0.833353i	$0.686418\pi$
$458$	0	0
$459$	−57.3457	−2.67667
$460$	0	0
$461$	6.63058	0.308817	0.154409	−	0.988007i	$-0.450653\pi$
$461$	6.63058	0.308817	0.154409	+	0.988007i	$0.450653\pi$
$462$	0	0
$463$	21.0220i	0.976975i	0.872571	+	0.488487i	$0.162451\pi$
$463$	21.0220i	0.976975i	−0.872571	+	0.488487i	$0.837549\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.8933i	0.735456i	0.929933	+	0.367728i	$0.119864\pi$
$467$	15.8933i	0.735456i	−0.929933	+	0.367728i	$0.880136\pi$
$468$	0	0
$469$	1.53956	0.0710905
$470$	0	0
$471$	21.9783	1.01271
$472$	0	0
$473$	− 33.1967i	− 1.52639i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	48.1006i	2.20237i
$478$	0	0
$479$	34.8564	1.59263	0.796315	−	0.604882i	$-0.206780\pi$
$479$	34.8564	1.59263	0.796315	+	0.604882i	$0.206780\pi$
$480$	0	0
$481$	12.0267	0.548371
$482$	0	0
$483$	− 8.52900i	− 0.388083i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 9.44393i	− 0.427945i	−0.976840	−	0.213973i	$-0.931360\pi$
$487$	− 9.44393i	− 0.427945i	0.976840	−	0.213973i	$-0.0686404\pi$
$488$	0	0
$489$	48.6944	2.20204
$490$	0	0
$491$	33.1577	1.49639	0.748193	−	0.663481i	$-0.230922\pi$
$491$	33.1577	1.49639	0.748193	+	0.663481i	$0.230922\pi$
$492$	0	0
$493$	− 33.3568i	− 1.50231i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.59667i	0.116477i
$498$	0	0
$499$	−12.7597	−0.571203	−0.285602	−	0.958348i	$-0.592193\pi$
$499$	−12.7597	−0.571203	−0.285602	+	0.958348i	$0.592193\pi$
$500$	0	0
$501$	−18.7712	−0.838633
$502$	0	0
$503$	17.0025i	0.758104i	0.925375	+	0.379052i	$0.123750\pi$
$503$	17.0025i	0.758104i	−0.925375	+	0.379052i	$0.876250\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	35.6359i	1.58265i
$508$	0	0
$509$	2.50950	0.111232	0.0556158	−	0.998452i	$-0.482288\pi$
$509$	2.50950	0.111232	0.0556158	+	0.998452i	$0.482288\pi$
$510$	0	0
$511$	−0.543345	−0.0240361
$512$	0	0
$513$	55.7707i	2.46234i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 41.3090i	− 1.81677i
$518$	0	0
$519$	−17.6093	−0.772964
$520$	0	0
$521$	13.1300	0.575237	0.287618	−	0.957745i	$-0.407136\pi$
$521$	13.1300	0.575237	0.287618	+	0.957745i	$0.407136\pi$
$522$	0	0
$523$	− 23.8301i	− 1.04202i	−0.853551	−	0.521010i	$-0.825556\pi$
$523$	− 23.8301i	− 1.04202i	0.853551	−	0.521010i	$-0.174444\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4.45350i	− 0.193998i
$528$	0	0
$529$	13.6023	0.591404
$530$	0	0
$531$	−22.9901	−0.997684
$532$	0	0
$533$	1.41500i	0.0612904i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 26.2544i	− 1.13296i
$538$	0	0
$539$	38.8726	1.67436
$540$	0	0
$541$	14.6995	0.631980	0.315990	−	0.948762i	$-0.397663\pi$
$541$	14.6995	0.631980	0.315990	+	0.948762i	$0.397663\pi$
$542$	0	0
$543$	− 29.9528i	− 1.28540i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	20.2209i	0.864584i	0.901734	+	0.432292i	$0.142295\pi$
$547$	20.2209i	0.864584i	−0.901734	+	0.432292i	$0.857705\pi$
$548$	0	0
$549$	−55.6084	−2.37331
$550$	0	0
$551$	−32.4406	−1.38202
$552$	0	0
$553$	− 12.2834i	− 0.522342i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 40.6784i	− 1.72360i	−0.507249	−	0.861799i	$-0.669338\pi$
$557$	− 40.6784i	− 1.72360i	0.507249	−	0.861799i	$-0.330662\pi$
$558$	0	0
$559$	−7.56787	−0.320087
$560$	0	0
$561$	79.1079	3.33994
$562$	0	0
$563$	19.4486i	0.819661i	0.912162	+	0.409831i	$0.134412\pi$
$563$	19.4486i	0.819661i	−0.912162	+	0.409831i	$0.865588\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 21.2453i	− 0.892217i
$568$	0	0
$569$	31.1850	1.30734	0.653672	−	0.756778i	$-0.273227\pi$
$569$	31.1850	1.30734	0.653672	+	0.756778i	$0.273227\pi$
$570$	0	0
$571$	−0.190921	−0.00798981	−0.00399491	−	0.999992i	$-0.501272\pi$
$571$	−0.190921	−0.00798981	−0.00399491	+	0.999992i	$0.501272\pi$
$572$	0	0
$573$	78.8623i	3.29452i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 29.0278i	− 1.20844i	−0.796816	−	0.604222i	$-0.793484\pi$
$577$	− 29.0278i	− 1.20844i	0.796816	−	0.604222i	$-0.206516\pi$
$578$	0	0
$579$	42.9772	1.78607
$580$	0	0
$581$	7.52441	0.312165
$582$	0	0
$583$	− 39.8106i	− 1.64879i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 11.5735i	− 0.477690i	−0.971058	−	0.238845i	$-0.923231\pi$
$587$	− 11.5735i	− 0.477690i	0.971058	−	0.238845i	$-0.0767687\pi$
$588$	0	0
$589$	−4.33118	−0.178463
$590$	0	0
$591$	58.6869	2.41405
$592$	0	0
$593$	− 34.5557i	− 1.41903i	−0.704689	−	0.709517i	$-0.748913\pi$
$593$	− 34.5557i	− 1.41903i	0.704689	−	0.709517i	$-0.251087\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 64.0688i	− 2.62216i
$598$	0	0
$599$	−26.9967	−1.10305	−0.551527	−	0.834157i	$-0.685955\pi$
$599$	−26.9967	−1.10305	−0.551527	+	0.834157i	$0.685955\pi$
$600$	0	0
$601$	47.8534	1.95198	0.975990	−	0.217816i	$-0.0698932\pi$
$601$	47.8534	1.95198	0.975990	+	0.217816i	$0.0698932\pi$
$602$	0	0
$603$	− 13.4469i	− 0.547601i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	12.3761i	0.502332i	0.967944	+	0.251166i	$0.0808140\pi$
$607$	12.3761i	0.502332i	−0.967944	+	0.251166i	$0.919186\pi$
$608$	0	0
$609$	23.5946	0.956100
$610$	0	0
$611$	−9.41722	−0.380980
$612$	0	0
$613$	− 12.9061i	− 0.521274i	−0.965437	−	0.260637i	$-0.916067\pi$
$613$	− 12.9061i	− 0.521274i	0.965437	−	0.260637i	$-0.0839325\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 40.5268i	− 1.63155i	−0.578372	−	0.815773i	$-0.696312\pi$
$617$	− 40.5268i	− 1.63155i	0.578372	−	0.815773i	$-0.303688\pi$
$618$	0	0
$619$	2.65889	0.106870	0.0534348	−	0.998571i	$-0.482983\pi$
$619$	2.65889	0.106870	0.0534348	+	0.998571i	$0.482983\pi$
$620$	0	0
$621$	−44.6944	−1.79353
$622$	0	0
$623$	− 6.77934i	− 0.271609i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 76.9351i	− 3.07249i
$628$	0	0
$629$	33.4310	1.33298
$630$	0	0
$631$	−0.139958	−0.00557165	−0.00278583	−	0.999996i	$-0.500887\pi$
$631$	−0.139958	−0.00557165	−0.00278583	+	0.999996i	$0.500887\pi$
$632$	0	0
$633$	4.98757i	0.198238i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 8.86180i	− 0.351117i
$638$	0	0
$639$	22.6800	0.897205
$640$	0	0
$641$	9.39675	0.371149	0.185575	−	0.982630i	$-0.440585\pi$
$641$	9.39675	0.371149	0.185575	+	0.982630i	$0.440585\pi$
$642$	0	0
$643$	− 6.02863i	− 0.237746i	−0.992909	−	0.118873i	$-0.962072\pi$
$643$	− 6.02863i	− 0.237746i	0.992909	−	0.118873i	$-0.0379281\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.3861i	0.683517i	0.939788	+	0.341758i	$0.111022\pi$
$647$	17.3861i	0.683517i	−0.939788	+	0.341758i	$0.888978\pi$
$648$	0	0
$649$	19.0278	0.746907
$650$	0	0
$651$	3.15014	0.123464
$652$	0	0
$653$	8.02832i	0.314173i	0.987585	+	0.157086i	$0.0502101\pi$
$653$	8.02832i	0.314173i	−0.987585	+	0.157086i	$0.949790\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.74570i	0.185147i
$658$	0	0
$659$	33.8809	1.31981	0.659907	−	0.751348i	$-0.270596\pi$
$659$	33.8809	1.31981	0.659907	+	0.751348i	$0.270596\pi$
$660$	0	0
$661$	34.3840	1.33738	0.668691	−	0.743541i	$-0.266855\pi$
$661$	34.3840	1.33738	0.668691	+	0.743541i	$0.266855\pi$
$662$	0	0
$663$	− 18.0343i	− 0.700393i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 25.9978i	− 1.00664i
$668$	0	0
$669$	−89.6191	−3.46487
$670$	0	0
$671$	46.0245	1.77676
$672$	0	0
$673$	− 10.3689i	− 0.399693i	−0.979827	−	0.199847i	$-0.935956\pi$
$673$	− 10.3689i	− 0.399693i	0.979827	−	0.199847i	$-0.0640444\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.0616i	1.07850i	0.842147	+	0.539248i	$0.181291\pi$
$677$	28.0616i	1.07850i	−0.842147	+	0.539248i	$0.818709\pi$
$678$	0	0
$679$	8.47175	0.325116
$680$	0	0
$681$	5.82070	0.223050
$682$	0	0
$683$	− 12.2209i	− 0.467621i	−0.972282	−	0.233810i	$-0.924881\pi$
$683$	− 12.2209i	− 0.467621i	0.972282	−	0.233810i	$-0.0751195\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 70.0317i	− 2.67188i
$688$	0	0
$689$	−9.07565	−0.345755
$690$	0	0
$691$	46.1797	1.75676	0.878379	−	0.477964i	$-0.158625\pi$
$691$	46.1797	1.75676	0.878379	+	0.477964i	$0.158625\pi$
$692$	0	0
$693$	39.9679i	1.51825i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.93332i	0.148985i
$698$	0	0
$699$	−59.3983	−2.24665
$700$	0	0
$701$	−46.8161	−1.76822	−0.884110	−	0.467279i	$-0.845234\pi$
$701$	−46.8161	−1.76822	−0.884110	+	0.467279i	$0.845234\pi$
$702$	0	0
$703$	− 32.5128i	− 1.22624i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.87955i	0.145905i
$708$	0	0
$709$	−42.0840	−1.58050	−0.790248	−	0.612787i	$-0.790048\pi$
$709$	−42.0840	−1.58050	−0.790248	+	0.612787i	$0.790048\pi$
$710$	0	0
$711$	−107.286	−4.02354
$712$	0	0
$713$	− 3.47100i	− 0.129990i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 38.7494i	− 1.44712i
$718$	0	0
$719$	−1.77685	−0.0662653	−0.0331327	−	0.999451i	$-0.510548\pi$
$719$	−1.77685	−0.0662653	−0.0331327	+	0.999451i	$0.510548\pi$
$720$	0	0
$721$	14.0934	0.524865
$722$	0	0
$723$	24.0267i	0.893563i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 17.4346i	− 0.646614i	−0.946294	−	0.323307i	$-0.895205\pi$
$727$	− 17.4346i	− 0.646614i	0.946294	−	0.323307i	$-0.104795\pi$
$728$	0	0
$729$	−43.8362	−1.62356
$730$	0	0
$731$	−21.0366	−0.778068
$732$	0	0
$733$	8.31827i	0.307242i	0.988130	+	0.153621i	$0.0490936\pi$
$733$	8.31827i	0.307242i	−0.988130	+	0.153621i	$0.950906\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.1294i	0.409957i
$738$	0	0
$739$	0.139958	0.00514845	0.00257422	−	0.999997i	$-0.499181\pi$
$739$	0.139958	0.00514845	0.00257422	+	0.999997i	$0.499181\pi$
$740$	0	0
$741$	−17.5389	−0.644309
$742$	0	0
$743$	3.92389i	0.143954i	0.997406	+	0.0719768i	$0.0229308\pi$
$743$	3.92389i	0.143954i	−0.997406	+	0.0719768i	$0.977069\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 65.7200i	− 2.40457i
$748$	0	0
$749$	−13.7934	−0.503998
$750$	0	0
$751$	38.2786	1.39681	0.698403	−	0.715705i	$-0.253894\pi$
$751$	38.2786	1.39681	0.698403	+	0.715705i	$0.253894\pi$
$752$	0	0
$753$	− 6.20538i	− 0.226137i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 31.0640i	− 1.12904i	−0.825420	−	0.564519i	$-0.809062\pi$
$757$	− 31.0640i	− 1.12904i	0.825420	−	0.564519i	$-0.190938\pi$
$758$	0	0
$759$	61.6556	2.23795
$760$	0	0
$761$	−2.83945	−0.102930	−0.0514649	−	0.998675i	$-0.516389\pi$
$761$	−2.83945	−0.102930	−0.0514649	+	0.998675i	$0.516389\pi$
$762$	0	0
$763$	− 4.81931i	− 0.174471i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 4.33778i	− 0.156628i
$768$	0	0
$769$	−38.7888	−1.39876	−0.699379	−	0.714751i	$-0.746540\pi$
$769$	−38.7888	−1.39876	−0.699379	+	0.714751i	$0.746540\pi$
$770$	0	0
$771$	−9.50839	−0.342436
$772$	0	0
$773$	− 25.7150i	− 0.924905i	−0.886644	−	0.462453i	$-0.846970\pi$
$773$	− 25.7150i	− 0.924905i	0.886644	−	0.462453i	$-0.153030\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	23.6471i	0.848335i
$778$	0	0
$779$	3.82529	0.137055
$780$	0	0
$781$	−18.7712	−0.671685
$782$	0	0
$783$	− 123.642i	− 4.41861i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	41.7817i	1.48936i	0.667423	+	0.744679i	$0.267397\pi$
$787$	41.7817i	1.48936i	−0.667423	+	0.744679i	$0.732603\pi$
$788$	0	0
$789$	−10.0673	−0.358406
$790$	0	0
$791$	−4.83552	−0.171931
$792$	0	0
$793$	− 10.4922i	− 0.372590i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	36.2818i	1.28517i	0.766216	+	0.642583i	$0.222137\pi$
$797$	36.2818i	1.28517i	−0.766216	+	0.642583i	$0.777863\pi$
$798$	0	0
$799$	−26.1774	−0.926088
$800$	0	0
$801$	−59.2124	−2.09217
$802$	0	0
$803$	− 3.92780i	− 0.138609i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	33.1967i	1.16858i
$808$	0	0
$809$	37.8556	1.33093	0.665466	−	0.746428i	$-0.268233\pi$
$809$	37.8556	1.33093	0.665466	+	0.746428i	$0.268233\pi$
$810$	0	0
$811$	−15.5751	−0.546915	−0.273457	−	0.961884i	$-0.588167\pi$
$811$	−15.5751	−0.546915	−0.273457	+	0.961884i	$0.588167\pi$
$812$	0	0
$813$	− 36.2877i	− 1.27267i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	20.4588i	0.715764i
$818$	0	0
$819$	9.11149	0.318381
$820$	0	0
$821$	−9.42283	−0.328859	−0.164430	−	0.986389i	$-0.552578\pi$
$821$	−9.42283	−0.328859	−0.164430	+	0.986389i	$0.552578\pi$
$822$	0	0
$823$	13.9636i	0.486741i	0.969933	+	0.243371i	$0.0782532\pi$
$823$	13.9636i	0.486741i	−0.969933	+	0.243371i	$0.921747\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19.6886i	0.684641i	0.939583	+	0.342320i	$0.111213\pi$
$827$	19.6886i	0.684641i	−0.939583	+	0.342320i	$0.888787\pi$
$828$	0	0
$829$	−52.9595	−1.83936	−0.919681	−	0.392668i	$-0.871552\pi$
$829$	−52.9595	−1.83936	−0.919681	+	0.392668i	$0.871552\pi$
$830$	0	0
$831$	−8.19266	−0.284200
$832$	0	0
$833$	− 24.6334i	− 0.853498i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 16.5076i	− 0.570587i
$838$	0	0
$839$	17.1392	0.591709	0.295855	−	0.955233i	$-0.404396\pi$
$839$	17.1392	0.591709	0.295855	+	0.955233i	$0.404396\pi$
$840$	0	0
$841$	42.9201	1.48000
$842$	0	0
$843$	− 76.3594i	− 2.62996i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 23.6347i	− 0.812097i
$848$	0	0
$849$	29.1590	1.00073
$850$	0	0
$851$	26.0556	0.893176
$852$	0	0
$853$	24.8256i	0.850011i	0.905191	+	0.425005i	$0.139728\pi$
$853$	24.8256i	0.850011i	−0.905191	+	0.425005i	$0.860272\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25.6584i	0.876474i	0.898859	+	0.438237i	$0.144397\pi$
$857$	25.6584i	0.876474i	−0.898859	+	0.438237i	$0.855603\pi$
$858$	0	0
$859$	−11.9818	−0.408812	−0.204406	−	0.978886i	$-0.565526\pi$
$859$	−11.9818	−0.408812	−0.204406	+	0.978886i	$0.565526\pi$
$860$	0	0
$861$	−2.78219	−0.0948169
$862$	0	0
$863$	33.9138i	1.15444i	0.816589	+	0.577220i	$0.195862\pi$
$863$	33.9138i	1.15444i	−0.816589	+	0.577220i	$0.804138\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	4.95441i	0.168261i
$868$	0	0
$869$	88.7956	3.01219
$870$	0	0
$871$	2.53717	0.0859688
$872$	0	0
$873$	− 73.9943i	− 2.50433i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 14.1871i	− 0.479066i	−0.970888	−	0.239533i	$-0.923006\pi$
$877$	− 14.1871i	− 0.479066i	0.970888	−	0.239533i	$-0.0769943\pi$
$878$	0	0
$879$	10.2950	0.347243
$880$	0	0
$881$	42.3201	1.42580	0.712901	−	0.701265i	$-0.247381\pi$
$881$	42.3201	1.42580	0.712901	+	0.701265i	$0.247381\pi$
$882$	0	0
$883$	− 51.7258i	− 1.74071i	−0.492422	−	0.870356i	$-0.663888\pi$
$883$	− 51.7258i	− 1.74071i	0.492422	−	0.870356i	$-0.336112\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30.1270i	1.01157i	0.862661	+	0.505783i	$0.168796\pi$
$887$	30.1270i	1.01157i	−0.862661	+	0.505783i	$0.831204\pi$
$888$	0	0
$889$	1.45773	0.0488907
$890$	0	0
$891$	153.580	5.14514
$892$	0	0
$893$	25.4584i	0.851932i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 14.0556i	− 0.469304i
$898$	0	0
$899$	9.60213	0.320249
$900$	0	0
$901$	−25.2279	−0.840462
$902$	0	0
$903$	− 14.8800i	− 0.495177i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 4.24848i	− 0.141068i	−0.997509	−	0.0705342i	$-0.977530\pi$
$907$	− 4.24848i	− 0.141068i	0.997509	−	0.0705342i	$-0.0224704\pi$
$908$	0	0
$909$	33.8849	1.12389
$910$	0	0
$911$	−49.1395	−1.62806	−0.814031	−	0.580821i	$-0.802732\pi$
$911$	−49.1395	−1.62806	−0.814031	+	0.580821i	$0.802732\pi$
$912$	0	0
$913$	54.3934i	1.80016i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.12988i	0.0703349i
$918$	0	0
$919$	−6.59412	−0.217520	−0.108760	−	0.994068i	$-0.534688\pi$
$919$	−6.59412	−0.217520	−0.108760	+	0.994068i	$0.534688\pi$
$920$	0	0
$921$	56.9768	1.87745
$922$	0	0
$923$	4.27927i	0.140854i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 123.095i	− 4.04297i
$928$	0	0
$929$	−54.3833	−1.78426	−0.892130	−	0.451779i	$-0.850789\pi$
$929$	−54.3833	−1.78426	−0.892130	+	0.451779i	$0.850789\pi$
$930$	0	0
$931$	−23.9568	−0.785154
$932$	0	0
$933$	5.04669i	0.165221i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 43.0888i	− 1.40765i	−0.710374	−	0.703825i	$-0.751474\pi$
$937$	− 43.0888i	− 1.40765i	0.710374	−	0.703825i	$-0.248526\pi$
$938$	0	0
$939$	66.5085	2.17042
$940$	0	0
$941$	12.6966	0.413899	0.206949	−	0.978352i	$-0.433646\pi$
$941$	12.6966	0.413899	0.206949	+	0.978352i	$0.433646\pi$
$942$	0	0
$943$	3.06557i	0.0998287i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29.6867i	0.964688i	0.875982	+	0.482344i	$0.160215\pi$
$947$	29.6867i	0.964688i	−0.875982	+	0.482344i	$0.839785\pi$
$948$	0	0
$949$	−0.895422	−0.0290666
$950$	0	0
$951$	−84.6019	−2.74341
$952$	0	0
$953$	− 7.62947i	− 0.247143i	−0.992336	−	0.123571i	$-0.960565\pi$
$953$	− 7.62947i	− 0.247143i	0.992336	−	0.123571i	$-0.0394348\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	170.563i	5.51353i
$958$	0	0
$959$	2.37266	0.0766171
$960$	0	0
$961$	−29.7180	−0.958645
$962$	0	0
$963$	120.475i	3.88224i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 37.0187i	− 1.19044i	−0.803562	−	0.595221i	$-0.797065\pi$
$967$	− 37.0187i	− 1.19044i	0.803562	−	0.595221i	$-0.202935\pi$
$968$	0	0
$969$	−48.7535	−1.56619
$970$	0	0
$971$	−17.2425	−0.553338	−0.276669	−	0.960965i	$-0.589231\pi$
$971$	−17.2425	−0.553338	−0.276669	+	0.960965i	$0.589231\pi$
$972$	0	0
$973$	7.72668i	0.247706i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 31.6112i	− 1.01133i	−0.862729	−	0.505666i	$-0.831247\pi$
$977$	− 31.6112i	− 1.01133i	0.862729	−	0.505666i	$-0.168753\pi$
$978$	0	0
$979$	49.0074	1.56628
$980$	0	0
$981$	−42.0930	−1.34393
$982$	0	0
$983$	− 51.8543i	− 1.65390i	−0.562278	−	0.826948i	$-0.690075\pi$
$983$	− 51.8543i	− 1.65390i	0.562278	−	0.826948i	$-0.309925\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 18.5163i	− 0.589380i
$988$	0	0
$989$	−16.3956	−0.521351
$990$	0	0
$991$	−37.7209	−1.19824	−0.599121	−	0.800658i	$-0.704483\pi$
$991$	−37.7209	−1.19824	−0.599121	+	0.800658i	$0.704483\pi$
$992$	0	0
$993$	11.9522i	0.379291i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 19.1882i	− 0.607698i	−0.952720	−	0.303849i	$-0.901728\pi$
$997$	− 19.1882i	− 0.607698i	0.952720	−	0.303849i	$-0.0982718\pi$
$998$	0	0
$999$	123.917	3.92058

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4100.2.d.c.1149.8		8
5.2	odd	4		4100.2.a.c.1.4		4
5.3	odd	4		164.2.a.a.1.1	✓	4
5.4	even	2	inner	4100.2.d.c.1149.1		8
15.8	even	4		1476.2.a.g.1.2		4
20.3	even	4		656.2.a.i.1.4		4
35.13	even	4		8036.2.a.i.1.4		4
40.3	even	4		2624.2.a.y.1.1		4
40.13	odd	4		2624.2.a.v.1.4		4
60.23	odd	4		5904.2.a.bp.1.2		4
205.163	odd	4		6724.2.a.c.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.a.a.1.1	✓	4	5.3	odd	4
656.2.a.i.1.4		4	20.3	even	4
1476.2.a.g.1.2		4	15.8	even	4
2624.2.a.v.1.4		4	40.13	odd	4
2624.2.a.y.1.1		4	40.3	even	4
4100.2.a.c.1.4		4	5.2	odd	4
4100.2.d.c.1149.1		8	5.4	even	2	inner
4100.2.d.c.1149.8		8	1.1	even	1	trivial
5904.2.a.bp.1.2		4	60.23	odd	4
6724.2.a.c.1.4		4	205.163	odd	4
8036.2.a.i.1.4		4	35.13	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:	\( N \)	\(=\)	\( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4100.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.24028i q^{3} -0.858626i q^{7} -7.49944 q^{9} +O(q^{10})\)	\(q+3.24028i q^{3} -0.858626i q^{7} -7.49944 q^{9} +6.20694 q^{11} -1.41500i q^{13} -3.93332i q^{17} -3.82529 q^{19} +2.78219 q^{21} -3.06557i q^{23} -14.5795i q^{27} +8.48057 q^{29} +1.13225 q^{31} +20.1123i q^{33} +8.49944i q^{37} +4.58500 q^{39} -1.00000 q^{41} -5.34832i q^{43} -6.65528i q^{47} +6.26276 q^{49} +12.7451 q^{51} -6.41389i q^{53} -12.3950i q^{57} +3.06557 q^{59} +7.41500 q^{61} +6.43922i q^{63} +1.79306i q^{67} +9.93332 q^{69} -3.02422 q^{71} -0.632807i q^{73} -5.32944i q^{77} +14.3059 q^{79} +24.7433 q^{81} +8.76332i q^{83} +27.4795i q^{87} +7.89557 q^{89} -1.21496 q^{91} +3.66882i q^{93} +9.86664i q^{97} -46.5486 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9}+O(q^{10}) \) 8 * q - 24 * q^9	\( 8 q - 24 q^{9} + 8 q^{11} - 12 q^{19} + 8 q^{29} - 16 q^{31} + 48 q^{39} - 8 q^{41} - 32 q^{49} - 8 q^{51} - 24 q^{59} + 48 q^{61} + 56 q^{69} - 4 q^{71} + 36 q^{79} + 56 q^{81} - 8 q^{89} + 72 q^{91} - 116 q^{99}+O(q^{100}) \) 8 * q - 24 * q^9 + 8 * q^11 - 12 * q^19 + 8 * q^29 - 16 * q^31 + 48 * q^39 - 8 * q^41 - 32 * q^49 - 8 * q^51 - 24 * q^59 + 48 * q^61 + 56 * q^69 - 4 * q^71 + 36 * q^79 + 56 * q^81 - 8 * q^89 + 72 * q^91 - 116 * q^99

Introduction

L-functions

Modular forms

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Fields

Representations

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Database

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