LMFDB - Embedded newform 4100.2.d.f.1149.1

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 27x^{12} + 281x^{10} + 1405x^{8} + 3397x^{6} + 3488x^{4} + 1281x^{2} + 121 $ x^14 + 27x^12 + 281x^10 + 1405x^8 + 3397x^6 + 3488x^4 + 1281x^2 + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1149.1
Root			$-2.88961i$ of defining polynomial
Character	$\chi$	$=$	4100.1149
Dual form			4100.2.d.f.1149.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.88961i q^{3} -4.40706i q^{7} -5.34984 q^{9} +O(q^{10})$	$q-2.88961i q^{3} -4.40706i q^{7} -5.34984 q^{9} +3.64440 q^{11} -0.559720i q^{13} +0.883584i q^{17} +2.97768 q^{19} -12.7347 q^{21} -9.05146i q^{23} +6.79012i q^{27} +3.10543 q^{29} -0.274514 q^{31} -10.5309i q^{33} +0.668636i q^{37} -1.61737 q^{39} -1.00000 q^{41} -9.53253i q^{43} +3.74043i q^{47} -12.4221 q^{49} +2.55321 q^{51} -9.11015i q^{53} -8.60433i q^{57} -10.5109 q^{59} -9.12149 q^{61} +23.5770i q^{63} +4.89074i q^{67} -26.1552 q^{69} +12.7884 q^{71} -6.11104i q^{73} -16.0611i q^{77} +11.8396 q^{79} +3.57127 q^{81} -0.381590i q^{83} -8.97349i q^{87} -7.57265 q^{89} -2.46672 q^{91} +0.793239i q^{93} +16.0284i q^{97} -19.4970 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 12 q^{9}+O(q^{10}) $ 14 * q - 12 * q^9	$ 14 q - 12 q^{9} - 6 q^{11} - 6 q^{19} - 26 q^{21} + 2 q^{29} - 8 q^{31} - 26 q^{39} - 14 q^{41} - 34 q^{49} - 42 q^{51} - 40 q^{59} - 20 q^{61} - 50 q^{69} - 46 q^{71} - 22 q^{79} - 26 q^{81} - 16 q^{89} - 38 q^{91} - 72 q^{99}+O(q^{100}) $ 14 * q - 12 * q^9 - 6 * q^11 - 6 * q^19 - 26 * q^21 + 2 * q^29 - 8 * q^31 - 26 * q^39 - 14 * q^41 - 34 * q^49 - 42 * q^51 - 40 * q^59 - 20 * q^61 - 50 * q^69 - 46 * q^71 - 22 * q^79 - 26 * q^81 - 16 * q^89 - 38 * q^91 - 72 * 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
$3$	− 2.88961i	− 1.66832i	−0.551525	−	0.834158i	$-0.685954\pi$
$3$	− 2.88961i	− 1.66832i	0.551525	−	0.834158i	$-0.314046\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.40706i	− 1.66571i	−0.553491	−	0.832855i	$-0.686705\pi$
$7$	− 4.40706i	− 1.66571i	0.553491	−	0.832855i	$-0.313295\pi$
$8$	0	0
$9$	−5.34984	−1.78328
$10$	0	0
$11$	3.64440	1.09883	0.549414	−	0.835550i	$-0.314851\pi$
$11$	3.64440	1.09883	0.549414	+	0.835550i	$0.314851\pi$
$12$	0	0
$13$	− 0.559720i	− 0.155238i	−0.996983	−	0.0776192i	$-0.975268\pi$
$13$	− 0.559720i	− 0.155238i	0.996983	−	0.0776192i	$-0.0247318\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.883584i	0.214301i	0.994243	+	0.107150i	$0.0341726\pi$
$17$	0.883584i	0.214301i	−0.994243	+	0.107150i	$0.965827\pi$
$18$	0	0
$19$	2.97768	0.683126	0.341563	−	0.939859i	$-0.389044\pi$
$19$	2.97768	0.683126	0.341563	+	0.939859i	$0.389044\pi$
$20$	0	0
$21$	−12.7347	−2.77893
$22$	0	0
$23$	− 9.05146i	− 1.88736i	−0.330861	−	0.943680i	$-0.607339\pi$
$23$	− 9.05146i	− 1.88736i	0.330861	−	0.943680i	$-0.392661\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	6.79012i	1.30676i
$28$	0	0
$29$	3.10543	0.576664	0.288332	−	0.957530i	$-0.406899\pi$
$29$	3.10543	0.576664	0.288332	+	0.957530i	$0.406899\pi$
$30$	0	0
$31$	−0.274514	−0.0493042	−0.0246521	−	0.999696i	$-0.507848\pi$
$31$	−0.274514	−0.0493042	−0.0246521	+	0.999696i	$0.507848\pi$
$32$	0	0
$33$	− 10.5309i	− 1.83319i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.668636i	0.109923i	0.998488	+	0.0549615i	$0.0175036\pi$
$37$	0.668636i	0.109923i	−0.998488	+	0.0549615i	$0.982496\pi$
$38$	0	0
$39$	−1.61737	−0.258987
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	− 9.53253i	− 1.45370i	−0.686797	−	0.726849i	$-0.740984\pi$
$43$	− 9.53253i	− 1.45370i	0.686797	−	0.726849i	$-0.259016\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.74043i	0.545598i	0.962071	+	0.272799i	$0.0879493\pi$
$47$	3.74043i	0.545598i	−0.962071	+	0.272799i	$0.912051\pi$
$48$	0	0
$49$	−12.4221	−1.77459
$50$	0	0
$51$	2.55321	0.357521
$52$	0	0
$53$	− 9.11015i	− 1.25137i	−0.780074	−	0.625687i	$-0.784819\pi$
$53$	− 9.11015i	− 1.25137i	0.780074	−	0.625687i	$-0.215181\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 8.60433i	− 1.13967i
$58$	0	0
$59$	−10.5109	−1.36841	−0.684204	−	0.729291i	$-0.739850\pi$
$59$	−10.5109	−1.36841	−0.684204	+	0.729291i	$0.739850\pi$
$60$	0	0
$61$	−9.12149	−1.16789	−0.583943	−	0.811794i	$-0.698491\pi$
$61$	−9.12149	−1.16789	−0.583943	+	0.811794i	$0.698491\pi$
$62$	0	0
$63$	23.5770i	2.97043i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.89074i	0.597499i	0.954332	+	0.298750i	$0.0965695\pi$
$67$	4.89074i	0.597499i	−0.954332	+	0.298750i	$0.903430\pi$
$68$	0	0
$69$	−26.1552	−3.14871
$70$	0	0
$71$	12.7884	1.51770	0.758852	−	0.651263i	$-0.225761\pi$
$71$	12.7884	1.51770	0.758852	+	0.651263i	$0.225761\pi$
$72$	0	0
$73$	− 6.11104i	− 0.715243i	−0.933867	−	0.357622i	$-0.883588\pi$
$73$	− 6.11104i	− 0.715243i	0.933867	−	0.357622i	$-0.116412\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 16.0611i	− 1.83033i
$78$	0	0
$79$	11.8396	1.33206	0.666029	−	0.745926i	$-0.267993\pi$
$79$	11.8396	1.33206	0.666029	+	0.745926i	$0.267993\pi$
$80$	0	0
$81$	3.57127	0.396808
$82$	0	0
$83$	− 0.381590i	− 0.0418850i	−0.999781	−	0.0209425i	$-0.993333\pi$
$83$	− 0.381590i	− 0.0418850i	0.999781	−	0.0209425i	$-0.00666669\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 8.97349i	− 0.962059i
$88$	0	0
$89$	−7.57265	−0.802700	−0.401350	−	0.915925i	$-0.631459\pi$
$89$	−7.57265	−0.802700	−0.401350	+	0.915925i	$0.631459\pi$
$90$	0	0
$91$	−2.46672	−0.258582
$92$	0	0
$93$	0.793239i	0.0822550i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.0284i	1.62744i	0.581260	+	0.813718i	$0.302560\pi$
$97$	16.0284i	1.62744i	−0.581260	+	0.813718i	$0.697440\pi$
$98$	0	0
$99$	−19.4970	−1.95952
$100$	0	0
$101$	3.56445	0.354676	0.177338	−	0.984150i	$-0.443251\pi$
$101$	3.56445	0.354676	0.177338	+	0.984150i	$0.443251\pi$
$102$	0	0
$103$	19.6724i	1.93838i	0.246316	+	0.969190i	$0.420780\pi$
$103$	19.6724i	1.93838i	−0.246316	+	0.969190i	$0.579220\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.699416i	0.0676151i	0.999428	+	0.0338075i	$0.0107633\pi$
$107$	0.699416i	0.0676151i	−0.999428	+	0.0338075i	$0.989237\pi$
$108$	0	0
$109$	18.2473	1.74777	0.873885	−	0.486133i	$-0.161593\pi$
$109$	18.2473	1.74777	0.873885	+	0.486133i	$0.161593\pi$
$110$	0	0
$111$	1.93210	0.183386
$112$	0	0
$113$	8.37177i	0.787550i	0.919207	+	0.393775i	$0.128831\pi$
$113$	8.37177i	0.787550i	−0.919207	+	0.393775i	$0.871169\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.99441i	0.276833i
$118$	0	0
$119$	3.89401	0.356963
$120$	0	0
$121$	2.28165	0.207423
$122$	0	0
$123$	2.88961i	0.260547i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 14.5280i	− 1.28915i	−0.764541	−	0.644575i	$-0.777034\pi$
$127$	− 14.5280i	− 1.28915i	0.764541	−	0.644575i	$-0.222966\pi$
$128$	0	0
$129$	−27.5453	−2.42523
$130$	0	0
$131$	4.10048	0.358260	0.179130	−	0.983825i	$-0.442672\pi$
$131$	4.10048	0.358260	0.179130	+	0.983825i	$0.442672\pi$
$132$	0	0
$133$	− 13.1228i	− 1.13789i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.53219i	0.814390i	0.913341	+	0.407195i	$0.133493\pi$
$137$	9.53219i	0.814390i	−0.913341	+	0.407195i	$0.866507\pi$
$138$	0	0
$139$	8.93171	0.757578	0.378789	−	0.925483i	$-0.376341\pi$
$139$	8.93171	0.757578	0.378789	+	0.925483i	$0.376341\pi$
$140$	0	0
$141$	10.8084	0.910230
$142$	0	0
$143$	− 2.03984i	− 0.170580i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	35.8951i	2.96058i
$148$	0	0
$149$	−20.7238	−1.69776	−0.848878	−	0.528588i	$-0.822722\pi$
$149$	−20.7238	−1.69776	−0.848878	+	0.528588i	$0.822722\pi$
$150$	0	0
$151$	−10.7764	−0.876967	−0.438484	−	0.898739i	$-0.644484\pi$
$151$	−10.7764	−0.876967	−0.438484	+	0.898739i	$0.644484\pi$
$152$	0	0
$153$	− 4.72703i	− 0.382158i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	11.5555i	0.922226i	0.887341	+	0.461113i	$0.152550\pi$
$157$	11.5555i	0.922226i	−0.887341	+	0.461113i	$0.847450\pi$
$158$	0	0
$159$	−26.3248	−2.08769
$160$	0	0
$161$	−39.8903	−3.14379
$162$	0	0
$163$	− 1.29570i	− 0.101487i	−0.998712	−	0.0507436i	$-0.983841\pi$
$163$	− 1.29570i	− 0.101487i	0.998712	−	0.0507436i	$-0.0161591\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.68365i	0.130285i	0.997876	+	0.0651423i	$0.0207501\pi$
$167$	1.68365i	0.130285i	−0.997876	+	0.0651423i	$0.979250\pi$
$168$	0	0
$169$	12.6867	0.975901
$170$	0	0
$171$	−15.9301	−1.21821
$172$	0	0
$173$	− 21.1503i	− 1.60803i	−0.594609	−	0.804015i	$-0.702693\pi$
$173$	− 21.1503i	− 1.60803i	0.594609	−	0.804015i	$-0.297307\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	30.3725i	2.28294i
$178$	0	0
$179$	14.2867	1.06784	0.533921	−	0.845534i	$-0.320718\pi$
$179$	14.2867	1.06784	0.533921	+	0.845534i	$0.320718\pi$
$180$	0	0
$181$	0.747123	0.0555332	0.0277666	−	0.999614i	$-0.491160\pi$
$181$	0.747123	0.0555332	0.0277666	+	0.999614i	$0.491160\pi$
$182$	0	0
$183$	26.3575i	1.94840i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.22013i	0.235480i
$188$	0	0
$189$	29.9244	2.17668
$190$	0	0
$191$	16.6227	1.20278	0.601389	−	0.798957i	$-0.294614\pi$
$191$	16.6227	1.20278	0.601389	+	0.798957i	$0.294614\pi$
$192$	0	0
$193$	4.18178i	0.301011i	0.988609	+	0.150505i	$0.0480901\pi$
$193$	4.18178i	0.301011i	−0.988609	+	0.150505i	$0.951910\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.1535i	1.50712i	0.657377	+	0.753562i	$0.271666\pi$
$197$	21.1535i	1.50712i	−0.657377	+	0.753562i	$0.728334\pi$
$198$	0	0
$199$	−11.0451	−0.782964	−0.391482	−	0.920186i	$-0.628037\pi$
$199$	−11.0451	−0.782964	−0.391482	+	0.920186i	$0.628037\pi$
$200$	0	0
$201$	14.1323	0.996818
$202$	0	0
$203$	− 13.6858i	− 0.960556i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	48.4238i	3.36569i
$208$	0	0
$209$	10.8519	0.750638
$210$	0	0
$211$	−7.84241	−0.539894	−0.269947	−	0.962875i	$-0.587006\pi$
$211$	−7.84241	−0.539894	−0.269947	+	0.962875i	$0.587006\pi$
$212$	0	0
$213$	− 36.9535i	− 2.53201i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.20980i	0.0821265i
$218$	0	0
$219$	−17.6585	−1.19325
$220$	0	0
$221$	0.494559	0.0332677
$222$	0	0
$223$	17.2477i	1.15499i	0.816394	+	0.577495i	$0.195970\pi$
$223$	17.2477i	1.15499i	−0.816394	+	0.577495i	$0.804030\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.08259i	0.602833i	0.953493	+	0.301416i	$0.0974594\pi$
$227$	9.08259i	0.602833i	−0.953493	+	0.301416i	$0.902541\pi$
$228$	0	0
$229$	18.2653	1.20701	0.603504	−	0.797360i	$-0.293771\pi$
$229$	18.2653	1.20701	0.603504	+	0.797360i	$0.293771\pi$
$230$	0	0
$231$	−46.4102	−3.05357
$232$	0	0
$233$	− 12.0003i	− 0.786165i	−0.919503	−	0.393082i	$-0.871409\pi$
$233$	− 12.0003i	− 0.786165i	0.919503	−	0.393082i	$-0.128591\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 34.2118i	− 2.22229i
$238$	0	0
$239$	−0.336792	−0.0217853	−0.0108926	−	0.999941i	$-0.503467\pi$
$239$	−0.336792	−0.0217853	−0.0108926	+	0.999941i	$0.503467\pi$
$240$	0	0
$241$	17.4750	1.12566	0.562832	−	0.826571i	$-0.309712\pi$
$241$	17.4750	1.12566	0.562832	+	0.826571i	$0.309712\pi$
$242$	0	0
$243$	10.0508i	0.644758i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.66667i	− 0.106047i
$248$	0	0
$249$	−1.10265	−0.0698774
$250$	0	0
$251$	−5.02355	−0.317084	−0.158542	−	0.987352i	$-0.550679\pi$
$251$	−5.02355	−0.317084	−0.158542	+	0.987352i	$0.550679\pi$
$252$	0	0
$253$	− 32.9871i	− 2.07388i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 17.9132i	− 1.11740i	−0.829371	−	0.558698i	$-0.811301\pi$
$257$	− 17.9132i	− 1.11740i	0.829371	−	0.558698i	$-0.188699\pi$
$258$	0	0
$259$	2.94672	0.183100
$260$	0	0
$261$	−16.6136	−1.02835
$262$	0	0
$263$	− 2.89167i	− 0.178308i	−0.996018	−	0.0891540i	$-0.971584\pi$
$263$	− 2.89167i	− 0.178308i	0.996018	−	0.0891540i	$-0.0284163\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	21.8820i	1.33916i
$268$	0	0
$269$	17.9294	1.09318	0.546588	−	0.837402i	$-0.315926\pi$
$269$	17.9294	1.09318	0.546588	+	0.837402i	$0.315926\pi$
$270$	0	0
$271$	3.07660	0.186890	0.0934451	−	0.995624i	$-0.470212\pi$
$271$	3.07660	0.186890	0.0934451	+	0.995624i	$0.470212\pi$
$272$	0	0
$273$	7.12785i	0.431397i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 16.7410i	− 1.00587i	−0.864325	−	0.502934i	$-0.832254\pi$
$277$	− 16.7410i	− 1.00587i	0.864325	−	0.502934i	$-0.167746\pi$
$278$	0	0
$279$	1.46861	0.0879232
$280$	0	0
$281$	0.813495	0.0485291	0.0242645	−	0.999706i	$-0.492276\pi$
$281$	0.813495	0.0485291	0.0242645	+	0.999706i	$0.492276\pi$
$282$	0	0
$283$	6.24731i	0.371364i	0.982610	+	0.185682i	$0.0594495\pi$
$283$	6.24731i	0.371364i	−0.982610	+	0.185682i	$0.940551\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.40706i	0.260140i
$288$	0	0
$289$	16.2193	0.954075
$290$	0	0
$291$	46.3158	2.71508
$292$	0	0
$293$	21.3806i	1.24907i	0.780998	+	0.624534i	$0.214711\pi$
$293$	21.3806i	1.24907i	−0.780998	+	0.624534i	$0.785289\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	24.7459i	1.43590i
$298$	0	0
$299$	−5.06628	−0.292990
$300$	0	0
$301$	−42.0104	−2.42144
$302$	0	0
$303$	− 10.2999i	− 0.591711i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	17.5423i	1.00119i	0.865681	+	0.500596i	$0.166886\pi$
$307$	17.5423i	1.00119i	−0.865681	+	0.500596i	$0.833114\pi$
$308$	0	0
$309$	56.8456	3.23383
$310$	0	0
$311$	−20.7551	−1.17691	−0.588456	−	0.808529i	$-0.700264\pi$
$311$	−20.7551	−1.17691	−0.588456	+	0.808529i	$0.700264\pi$
$312$	0	0
$313$	8.13300i	0.459704i	0.973226	+	0.229852i	$0.0738243\pi$
$313$	8.13300i	0.459704i	−0.973226	+	0.229852i	$0.926176\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 10.1431i	− 0.569694i	−0.958573	−	0.284847i	$-0.908057\pi$
$317$	− 10.1431i	− 0.569694i	0.958573	−	0.284847i	$-0.0919428\pi$
$318$	0	0
$319$	11.3174	0.633655
$320$	0	0
$321$	2.02104	0.112803
$322$	0	0
$323$	2.63103i	0.146394i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 52.7274i	− 2.91583i
$328$	0	0
$329$	16.4843	0.908808
$330$	0	0
$331$	−8.15799	−0.448404	−0.224202	−	0.974543i	$-0.571977\pi$
$331$	−8.15799	−0.448404	−0.224202	+	0.974543i	$0.571977\pi$
$332$	0	0
$333$	− 3.57710i	− 0.196024i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	27.0916i	1.47577i	0.674925	+	0.737886i	$0.264176\pi$
$337$	27.0916i	1.47577i	−0.674925	+	0.737886i	$0.735824\pi$
$338$	0	0
$339$	24.1911	1.31388
$340$	0	0
$341$	−1.00044	−0.0541768
$342$	0	0
$343$	23.8957i	1.29025i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 33.6631i	− 1.80713i	−0.428450	−	0.903566i	$-0.640940\pi$
$347$	− 33.6631i	− 1.80713i	0.428450	−	0.903566i	$-0.359060\pi$
$348$	0	0
$349$	−5.14985	−0.275665	−0.137833	−	0.990456i	$-0.544014\pi$
$349$	−5.14985	−0.275665	−0.137833	+	0.990456i	$0.544014\pi$
$350$	0	0
$351$	3.80056	0.202859
$352$	0	0
$353$	− 12.8798i	− 0.685520i	−0.939423	−	0.342760i	$-0.888638\pi$
$353$	− 12.8798i	− 0.685520i	0.939423	−	0.342760i	$-0.111362\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 11.2522i	− 0.595527i
$358$	0	0
$359$	−18.3809	−0.970106	−0.485053	−	0.874485i	$-0.661200\pi$
$359$	−18.3809	−0.970106	−0.485053	+	0.874485i	$0.661200\pi$
$360$	0	0
$361$	−10.1334	−0.533338
$362$	0	0
$363$	− 6.59308i	− 0.346047i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 25.3371i	− 1.32258i	−0.750129	−	0.661292i	$-0.770008\pi$
$367$	− 25.3371i	− 1.32258i	0.750129	−	0.661292i	$-0.229992\pi$
$368$	0	0
$369$	5.34984	0.278502
$370$	0	0
$371$	−40.1489	−2.08443
$372$	0	0
$373$	38.0776i	1.97158i	0.167972	+	0.985792i	$0.446278\pi$
$373$	38.0776i	1.97158i	−0.167972	+	0.985792i	$0.553722\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1.73817i	− 0.0895204i
$378$	0	0
$379$	−13.7126	−0.704370	−0.352185	−	0.935930i	$-0.614561\pi$
$379$	−13.7126	−0.704370	−0.352185	+	0.935930i	$0.614561\pi$
$380$	0	0
$381$	−41.9802	−2.15071
$382$	0	0
$383$	26.2958i	1.34365i	0.740708	+	0.671827i	$0.234490\pi$
$383$	26.2958i	1.34365i	−0.740708	+	0.671827i	$0.765510\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	50.9975i	2.59235i
$388$	0	0
$389$	2.92383	0.148244	0.0741220	−	0.997249i	$-0.476385\pi$
$389$	2.92383	0.148244	0.0741220	+	0.997249i	$0.476385\pi$
$390$	0	0
$391$	7.99772	0.404462
$392$	0	0
$393$	− 11.8488i	− 0.597691i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	36.7908i	1.84648i	0.384226	+	0.923239i	$0.374468\pi$
$397$	36.7908i	1.84648i	−0.384226	+	0.923239i	$0.625532\pi$
$398$	0	0
$399$	−37.9198	−1.89836
$400$	0	0
$401$	−37.9513	−1.89520	−0.947598	−	0.319465i	$-0.896497\pi$
$401$	−37.9513	−1.89520	−0.947598	+	0.319465i	$0.896497\pi$
$402$	0	0
$403$	0.153651i	0.00765390i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.43678i	0.120787i
$408$	0	0
$409$	−2.15425	−0.106521	−0.0532604	−	0.998581i	$-0.516961\pi$
$409$	−2.15425	−0.106521	−0.0532604	+	0.998581i	$0.516961\pi$
$410$	0	0
$411$	27.5443	1.35866
$412$	0	0
$413$	46.3223i	2.27937i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 25.8092i	− 1.26388i
$418$	0	0
$419$	−39.2760	−1.91876	−0.959378	−	0.282123i	$-0.908962\pi$
$419$	−39.2760	−1.91876	−0.959378	+	0.282123i	$0.908962\pi$
$420$	0	0
$421$	8.19123	0.399216	0.199608	−	0.979876i	$-0.436033\pi$
$421$	8.19123	0.399216	0.199608	+	0.979876i	$0.436033\pi$
$422$	0	0
$423$	− 20.0107i	− 0.972953i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	40.1989i	1.94536i
$428$	0	0
$429$	−5.89435	−0.284582
$430$	0	0
$431$	1.22961	0.0592281	0.0296140	−	0.999561i	$-0.490572\pi$
$431$	1.22961	0.0592281	0.0296140	+	0.999561i	$0.490572\pi$
$432$	0	0
$433$	− 16.4970i	− 0.792796i	−0.918079	−	0.396398i	$-0.870260\pi$
$433$	− 16.4970i	− 0.792796i	0.918079	−	0.396398i	$-0.129740\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 26.9523i	− 1.28930i
$438$	0	0
$439$	35.4067	1.68987	0.844934	−	0.534871i	$-0.179640\pi$
$439$	35.4067	1.68987	0.844934	+	0.534871i	$0.179640\pi$
$440$	0	0
$441$	66.4565	3.16459
$442$	0	0
$443$	− 30.9248i	− 1.46928i	−0.678455	−	0.734642i	$-0.737350\pi$
$443$	− 30.9248i	− 1.46928i	0.678455	−	0.734642i	$-0.262650\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	59.8836i	2.83240i
$448$	0	0
$449$	−11.4301	−0.539422	−0.269711	−	0.962941i	$-0.586928\pi$
$449$	−11.4301	−0.539422	−0.269711	+	0.962941i	$0.586928\pi$
$450$	0	0
$451$	−3.64440	−0.171608
$452$	0	0
$453$	31.1394i	1.46306i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.01813i	0.141182i	0.997505	+	0.0705912i	$0.0224886\pi$
$457$	3.01813i	0.141182i	−0.997505	+	0.0705912i	$0.977511\pi$
$458$	0	0
$459$	−5.99964	−0.280039
$460$	0	0
$461$	−0.0892765	−0.00415802	−0.00207901	−	0.999998i	$-0.500662\pi$
$461$	−0.0892765	−0.00415802	−0.00207901	+	0.999998i	$0.500662\pi$
$462$	0	0
$463$	− 31.5601i	− 1.46672i	−0.679840	−	0.733361i	$-0.737951\pi$
$463$	− 31.5601i	− 1.46672i	0.679840	−	0.733361i	$-0.262049\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 8.13497i	− 0.376442i	−0.982127	−	0.188221i	$-0.939728\pi$
$467$	− 8.13497i	− 0.376442i	0.982127	−	0.188221i	$-0.0602721\pi$
$468$	0	0
$469$	21.5538	0.995261
$470$	0	0
$471$	33.3908	1.53856
$472$	0	0
$473$	− 34.7404i	− 1.59736i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	48.7378i	2.23155i
$478$	0	0
$479$	0.0483785	0.00221047	0.00110523	−	0.999999i	$-0.499648\pi$
$479$	0.0483785	0.00221047	0.00110523	+	0.999999i	$0.499648\pi$
$480$	0	0
$481$	0.374249	0.0170643
$482$	0	0
$483$	115.267i	5.24484i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 10.6739i	− 0.483682i	−0.970316	−	0.241841i	$-0.922249\pi$
$487$	− 10.6739i	− 0.483682i	0.970316	−	0.241841i	$-0.0777512\pi$
$488$	0	0
$489$	−3.74407	−0.169313
$490$	0	0
$491$	−40.7366	−1.83842	−0.919208	−	0.393772i	$-0.871170\pi$
$491$	−40.7366	−1.83842	−0.919208	+	0.393772i	$0.871170\pi$
$492$	0	0
$493$	2.74391i	0.123580i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 56.3592i	− 2.52806i
$498$	0	0
$499$	7.84264	0.351085	0.175542	−	0.984472i	$-0.443832\pi$
$499$	7.84264	0.351085	0.175542	+	0.984472i	$0.443832\pi$
$500$	0	0
$501$	4.86509	0.217356
$502$	0	0
$503$	28.2531i	1.25974i	0.776699	+	0.629872i	$0.216893\pi$
$503$	28.2531i	1.25974i	−0.776699	+	0.629872i	$0.783107\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 36.6596i	− 1.62811i
$508$	0	0
$509$	31.2120	1.38345	0.691724	−	0.722162i	$-0.256852\pi$
$509$	31.2120	1.38345	0.691724	+	0.722162i	$0.256852\pi$
$510$	0	0
$511$	−26.9317	−1.19139
$512$	0	0
$513$	20.2188i	0.892682i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	13.6316i	0.599518i
$518$	0	0
$519$	−61.1162	−2.68270
$520$	0	0
$521$	36.3036	1.59049	0.795245	−	0.606289i	$-0.207342\pi$
$521$	36.3036	1.59049	0.795245	+	0.606289i	$0.207342\pi$
$522$	0	0
$523$	− 28.4639i	− 1.24464i	−0.782763	−	0.622319i	$-0.786190\pi$
$523$	− 28.4639i	− 1.24464i	0.782763	−	0.622319i	$-0.213810\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 0.242556i	− 0.0105659i
$528$	0	0
$529$	−58.9289	−2.56212
$530$	0	0
$531$	56.2319	2.44025
$532$	0	0
$533$	0.559720i	0.0242442i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 41.2831i	− 1.78150i
$538$	0	0
$539$	−45.2713	−1.94997
$540$	0	0
$541$	−40.2209	−1.72923	−0.864616	−	0.502433i	$-0.832438\pi$
$541$	−40.2209	−1.72923	−0.864616	+	0.502433i	$0.832438\pi$
$542$	0	0
$543$	− 2.15889i	− 0.0926470i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 24.0856i	− 1.02982i	−0.857243	−	0.514912i	$-0.827825\pi$
$547$	− 24.0856i	− 1.02982i	0.857243	−	0.514912i	$-0.172175\pi$
$548$	0	0
$549$	48.7985	2.08267
$550$	0	0
$551$	9.24698	0.393935
$552$	0	0
$553$	− 52.1777i	− 2.21882i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.15929i	0.260977i	0.991450	+	0.130489i	$0.0416546\pi$
$557$	6.15929i	0.260977i	−0.991450	+	0.130489i	$0.958345\pi$
$558$	0	0
$559$	−5.33555	−0.225670
$560$	0	0
$561$	9.30493	0.392854
$562$	0	0
$563$	13.5745i	0.572096i	0.958215	+	0.286048i	$0.0923416\pi$
$563$	13.5745i	0.572096i	−0.958215	+	0.286048i	$0.907658\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 15.7388i	− 0.660968i
$568$	0	0
$569$	5.73796	0.240548	0.120274	−	0.992741i	$-0.461623\pi$
$569$	5.73796	0.240548	0.120274	+	0.992741i	$0.461623\pi$
$570$	0	0
$571$	2.48825	0.104130	0.0520650	−	0.998644i	$-0.483420\pi$
$571$	2.48825	0.104130	0.0520650	+	0.998644i	$0.483420\pi$
$572$	0	0
$573$	− 48.0331i	− 2.00661i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 32.5095i	− 1.35339i	−0.736264	−	0.676694i	$-0.763412\pi$
$577$	− 32.5095i	− 1.35339i	0.736264	−	0.676694i	$-0.236588\pi$
$578$	0	0
$579$	12.0837	0.502181
$580$	0	0
$581$	−1.68169	−0.0697683
$582$	0	0
$583$	− 33.2010i	− 1.37505i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 18.1378i	− 0.748626i	−0.927302	−	0.374313i	$-0.877879\pi$
$587$	− 18.1378i	− 0.748626i	0.927302	−	0.374313i	$-0.122121\pi$
$588$	0	0
$589$	−0.817415	−0.0336810
$590$	0	0
$591$	61.1253	2.51436
$592$	0	0
$593$	− 3.79438i	− 0.155816i	−0.996961	−	0.0779082i	$-0.975176\pi$
$593$	− 3.79438i	− 0.155816i	0.996961	−	0.0779082i	$-0.0248241\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	31.9159i	1.30623i
$598$	0	0
$599$	2.64975	0.108266	0.0541329	−	0.998534i	$-0.482761\pi$
$599$	2.64975	0.108266	0.0541329	+	0.998534i	$0.482761\pi$
$600$	0	0
$601$	41.2659	1.68327	0.841636	−	0.540045i	$-0.181593\pi$
$601$	41.2659	1.68327	0.841636	+	0.540045i	$0.181593\pi$
$602$	0	0
$603$	− 26.1647i	− 1.06551i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 16.7828i	− 0.681192i	−0.940210	−	0.340596i	$-0.889371\pi$
$607$	− 16.7828i	− 0.681192i	0.940210	−	0.340596i	$-0.110629\pi$
$608$	0	0
$609$	−39.5467	−1.60251
$610$	0	0
$611$	2.09359	0.0846976
$612$	0	0
$613$	− 22.9738i	− 0.927905i	−0.885860	−	0.463952i	$-0.846431\pi$
$613$	− 22.9738i	− 0.927905i	0.885860	−	0.463952i	$-0.153569\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	43.3178i	1.74391i	0.489585	+	0.871955i	$0.337148\pi$
$617$	43.3178i	1.74391i	−0.489585	+	0.871955i	$0.662852\pi$
$618$	0	0
$619$	43.5406	1.75004	0.875022	−	0.484084i	$-0.160847\pi$
$619$	43.5406	1.75004	0.875022	+	0.484084i	$0.160847\pi$
$620$	0	0
$621$	61.4605	2.46632
$622$	0	0
$623$	33.3731i	1.33707i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 31.3576i	− 1.25230i
$628$	0	0
$629$	−0.590796	−0.0235566
$630$	0	0
$631$	39.2543	1.56269	0.781344	−	0.624101i	$-0.214535\pi$
$631$	39.2543	1.56269	0.781344	+	0.624101i	$0.214535\pi$
$632$	0	0
$633$	22.6615i	0.900714i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.95292i	0.275485i
$638$	0	0
$639$	−68.4159	−2.70649
$640$	0	0
$641$	−33.2309	−1.31254	−0.656272	−	0.754525i	$-0.727868\pi$
$641$	−33.2309	−1.31254	−0.656272	+	0.754525i	$0.727868\pi$
$642$	0	0
$643$	24.6583i	0.972427i	0.873840	+	0.486213i	$0.161622\pi$
$643$	24.6583i	0.972427i	−0.873840	+	0.486213i	$0.838378\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0.0460152i	0.00180904i	1.00000		0.000904522i	$0.000287918\pi$
$647$	0.0460152i	0.00180904i	−1.00000		0.000904522i	$0.999712\pi$
$648$	0	0
$649$	−38.3061	−1.50364
$650$	0	0
$651$	3.49585	0.137013
$652$	0	0
$653$	− 18.4658i	− 0.722621i	−0.932446	−	0.361310i	$-0.882330\pi$
$653$	− 18.4658i	− 0.722621i	0.932446	−	0.361310i	$-0.117670\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	32.6931i	1.27548i
$658$	0	0
$659$	27.5971	1.07503	0.537515	−	0.843254i	$-0.319363\pi$
$659$	27.5971	1.07503	0.537515	+	0.843254i	$0.319363\pi$
$660$	0	0
$661$	10.3521	0.402651	0.201325	−	0.979524i	$-0.435475\pi$
$661$	10.3521	0.402651	0.201325	+	0.979524i	$0.435475\pi$
$662$	0	0
$663$	− 1.42908i	− 0.0555010i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 28.1087i	− 1.08837i
$668$	0	0
$669$	49.8390	1.92689
$670$	0	0
$671$	−33.2423	−1.28331
$672$	0	0
$673$	− 47.6256i	− 1.83583i	−0.396776	−	0.917915i	$-0.629871\pi$
$673$	− 47.6256i	− 1.83583i	0.396776	−	0.917915i	$-0.370129\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 28.6435i	− 1.10086i	−0.834881	−	0.550430i	$-0.814464\pi$
$677$	− 28.6435i	− 1.10086i	0.834881	−	0.550430i	$-0.185536\pi$
$678$	0	0
$679$	70.6380	2.71084
$680$	0	0
$681$	26.2451	1.00572
$682$	0	0
$683$	22.8158i	0.873024i	0.899698	+	0.436512i	$0.143786\pi$
$683$	22.8158i	0.873024i	−0.899698	+	0.436512i	$0.856214\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 52.7797i	− 2.01367i
$688$	0	0
$689$	−5.09913	−0.194261
$690$	0	0
$691$	−22.6028	−0.859853	−0.429926	−	0.902864i	$-0.641460\pi$
$691$	−22.6028	−0.859853	−0.429926	+	0.902864i	$0.641460\pi$
$692$	0	0
$693$	85.9242i	3.26399i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 0.883584i	− 0.0334681i
$698$	0	0
$699$	−34.6761	−1.31157
$700$	0	0
$701$	4.41876	0.166894	0.0834471	−	0.996512i	$-0.473407\pi$
$701$	4.41876	0.166894	0.0834471	+	0.996512i	$0.473407\pi$
$702$	0	0
$703$	1.99098i	0.0750913i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 15.7087i	− 0.590787i
$708$	0	0
$709$	−4.49151	−0.168682	−0.0843410	−	0.996437i	$-0.526879\pi$
$709$	−4.49151	−0.168682	−0.0843410	+	0.996437i	$0.526879\pi$
$710$	0	0
$711$	−63.3399	−2.37543
$712$	0	0
$713$	2.48475i	0.0930547i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.973198i	0.0363447i
$718$	0	0
$719$	32.9583	1.22914	0.614568	−	0.788864i	$-0.289330\pi$
$719$	32.9583	1.22914	0.614568	+	0.788864i	$0.289330\pi$
$720$	0	0
$721$	86.6974	3.22878
$722$	0	0
$723$	− 50.4959i	− 1.87796i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	26.9760i	1.00048i	0.865886	+	0.500242i	$0.166756\pi$
$727$	26.9760i	1.00048i	−0.865886	+	0.500242i	$0.833244\pi$
$728$	0	0
$729$	39.7566	1.47247
$730$	0	0
$731$	8.42280	0.311528
$732$	0	0
$733$	− 24.0072i	− 0.886728i	−0.896342	−	0.443364i	$-0.853785\pi$
$733$	− 24.0072i	− 0.886728i	0.896342	−	0.443364i	$-0.146215\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	17.8238i	0.656549i
$738$	0	0
$739$	16.1225	0.593077	0.296538	−	0.955021i	$-0.404168\pi$
$739$	16.1225	0.593077	0.296538	+	0.955021i	$0.404168\pi$
$740$	0	0
$741$	−4.81601	−0.176921
$742$	0	0
$743$	− 2.51298i	− 0.0921923i	−0.998937	−	0.0460961i	$-0.985322\pi$
$743$	− 2.51298i	− 0.0921923i	0.998937	−	0.0460961i	$-0.0146781\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.04145i	0.0746926i
$748$	0	0
$749$	3.08236	0.112627
$750$	0	0
$751$	−29.0556	−1.06026	−0.530128	−	0.847918i	$-0.677856\pi$
$751$	−29.0556	−1.06026	−0.530128	+	0.847918i	$0.677856\pi$
$752$	0	0
$753$	14.5161i	0.528996i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 48.7412i	− 1.77153i	−0.464133	−	0.885765i	$-0.653634\pi$
$757$	− 48.7412i	− 1.77153i	0.464133	−	0.885765i	$-0.346366\pi$
$758$	0	0
$759$	−95.3199	−3.45989
$760$	0	0
$761$	9.60990	0.348359	0.174179	−	0.984714i	$-0.444273\pi$
$761$	9.60990	0.348359	0.174179	+	0.984714i	$0.444273\pi$
$762$	0	0
$763$	− 80.4167i	− 2.91128i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.88318i	0.212429i
$768$	0	0
$769$	−27.5111	−0.992076	−0.496038	−	0.868301i	$-0.665212\pi$
$769$	−27.5111	−0.992076	−0.496038	+	0.868301i	$0.665212\pi$
$770$	0	0
$771$	−51.7622	−1.86417
$772$	0	0
$773$	12.0120i	0.432040i	0.976389	+	0.216020i	$0.0693076\pi$
$773$	12.0120i	0.432040i	−0.976389	+	0.216020i	$0.930692\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 8.51486i	− 0.305469i
$778$	0	0
$779$	−2.97768	−0.106686
$780$	0	0
$781$	46.6061	1.66770
$782$	0	0
$783$	21.0863i	0.753562i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 38.6537i	− 1.37786i	−0.724830	−	0.688928i	$-0.758082\pi$
$787$	− 38.6537i	− 1.37786i	0.724830	−	0.688928i	$-0.241918\pi$
$788$	0	0
$789$	−8.35579	−0.297474
$790$	0	0
$791$	36.8948	1.31183
$792$	0	0
$793$	5.10547i	0.181301i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	43.1642i	1.52896i	0.644650	+	0.764478i	$0.277003\pi$
$797$	43.1642i	1.52896i	−0.644650	+	0.764478i	$0.722997\pi$
$798$	0	0
$799$	−3.30498	−0.116922
$800$	0	0
$801$	40.5125	1.43144
$802$	0	0
$803$	− 22.2711i	− 0.785929i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 51.8090i	− 1.82376i
$808$	0	0
$809$	−18.4675	−0.649283	−0.324641	−	0.945837i	$-0.605244\pi$
$809$	−18.4675	−0.649283	−0.324641	+	0.945837i	$0.605244\pi$
$810$	0	0
$811$	−0.929457	−0.0326377	−0.0163188	−	0.999867i	$-0.505195\pi$
$811$	−0.929457	−0.0326377	−0.0163188	+	0.999867i	$0.505195\pi$
$812$	0	0
$813$	− 8.89017i	− 0.311792i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 28.3848i	− 0.993059i
$818$	0	0
$819$	13.1965	0.461124
$820$	0	0
$821$	3.17549	0.110826	0.0554128	−	0.998464i	$-0.482353\pi$
$821$	3.17549	0.110826	0.0554128	+	0.998464i	$0.482353\pi$
$822$	0	0
$823$	− 35.8491i	− 1.24962i	−0.780777	−	0.624810i	$-0.785176\pi$
$823$	− 35.8491i	− 1.24962i	0.780777	−	0.624810i	$-0.214824\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.37374i	0.117317i	0.998278	+	0.0586583i	$0.0186822\pi$
$827$	3.37374i	0.117317i	−0.998278	+	0.0586583i	$0.981318\pi$
$828$	0	0
$829$	56.8926	1.97596	0.987980	−	0.154581i	$-0.0494028\pi$
$829$	56.8926	1.97596	0.987980	+	0.154581i	$0.0494028\pi$
$830$	0	0
$831$	−48.3749	−1.67811
$832$	0	0
$833$	− 10.9760i	− 0.380296i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.86398i	− 0.0644287i
$838$	0	0
$839$	−48.4811	−1.67375	−0.836877	−	0.547390i	$-0.815621\pi$
$839$	−48.4811	−1.67375	−0.836877	+	0.547390i	$0.815621\pi$
$840$	0	0
$841$	−19.3563	−0.667458
$842$	0	0
$843$	− 2.35068i	− 0.0809618i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 10.0554i	− 0.345507i
$848$	0	0
$849$	18.0523	0.619553
$850$	0	0
$851$	6.05213	0.207464
$852$	0	0
$853$	8.75342i	0.299711i	0.988708	+	0.149856i	$0.0478809\pi$
$853$	8.75342i	0.299711i	−0.988708	+	0.149856i	$0.952119\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 48.3660i	− 1.65215i	−0.563559	−	0.826076i	$-0.690568\pi$
$857$	− 48.3660i	− 1.65215i	0.563559	−	0.826076i	$-0.309432\pi$
$858$	0	0
$859$	34.2758	1.16947	0.584737	−	0.811223i	$-0.301198\pi$
$859$	34.2758	1.16947	0.584737	+	0.811223i	$0.301198\pi$
$860$	0	0
$861$	12.7347	0.433996
$862$	0	0
$863$	− 19.9090i	− 0.677709i	−0.940839	−	0.338855i	$-0.889961\pi$
$863$	− 19.9090i	− 0.677709i	0.940839	−	0.338855i	$-0.110039\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 46.8674i	− 1.59170i
$868$	0	0
$869$	43.1482	1.46370
$870$	0	0
$871$	2.73744	0.0927548
$872$	0	0
$873$	− 85.7493i	− 2.90217i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	31.1671i	1.05244i	0.850349	+	0.526219i	$0.176391\pi$
$877$	31.1671i	1.05244i	−0.850349	+	0.526219i	$0.823609\pi$
$878$	0	0
$879$	61.7816	2.08384
$880$	0	0
$881$	0.521369	0.0175654	0.00878269	−	0.999961i	$-0.497204\pi$
$881$	0.521369	0.0175654	0.00878269	+	0.999961i	$0.497204\pi$
$882$	0	0
$883$	16.3226i	0.549299i	0.961544	+	0.274649i	$0.0885617\pi$
$883$	16.3226i	0.549299i	−0.961544	+	0.274649i	$0.911438\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 20.0237i	− 0.672330i	−0.941803	−	0.336165i	$-0.890870\pi$
$887$	− 20.0237i	− 0.672330i	0.941803	−	0.336165i	$-0.109130\pi$
$888$	0	0
$889$	−64.0256	−2.14735
$890$	0	0
$891$	13.0151	0.436024
$892$	0	0
$893$	11.1378i	0.372712i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	14.6396i	0.488801i
$898$	0	0
$899$	−0.852485	−0.0284320
$900$	0	0
$901$	8.04958	0.268170
$902$	0	0
$903$	121.394i	4.03973i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 31.6570i	− 1.05115i	−0.850746	−	0.525577i	$-0.823850\pi$
$907$	− 31.6570i	− 1.05115i	0.850746	−	0.525577i	$-0.176150\pi$
$908$	0	0
$909$	−19.0692	−0.632486
$910$	0	0
$911$	8.41195	0.278700	0.139350	−	0.990243i	$-0.455499\pi$
$911$	8.41195	0.278700	0.139350	+	0.990243i	$0.455499\pi$
$912$	0	0
$913$	− 1.39067i	− 0.0460244i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 18.0710i	− 0.596758i
$918$	0	0
$919$	11.1387	0.367433	0.183716	−	0.982979i	$-0.441187\pi$
$919$	11.1387	0.367433	0.183716	+	0.982979i	$0.441187\pi$
$920$	0	0
$921$	50.6904	1.67030
$922$	0	0
$923$	− 7.15792i	− 0.235606i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 105.244i	− 3.45667i
$928$	0	0
$929$	3.36989	0.110563	0.0552813	−	0.998471i	$-0.482394\pi$
$929$	3.36989	0.110563	0.0552813	+	0.998471i	$0.482394\pi$
$930$	0	0
$931$	−36.9892	−1.21227
$932$	0	0
$933$	59.9740i	1.96346i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 58.2430i	− 1.90272i	−0.308087	−	0.951358i	$-0.599689\pi$
$937$	− 58.2430i	− 1.90272i	0.308087	−	0.951358i	$-0.400311\pi$
$938$	0	0
$939$	23.5012	0.766932
$940$	0	0
$941$	4.61565	0.150466	0.0752330	−	0.997166i	$-0.476030\pi$
$941$	4.61565	0.150466	0.0752330	+	0.997166i	$0.476030\pi$
$942$	0	0
$943$	9.05146i	0.294756i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	27.0652i	0.879502i	0.898120	+	0.439751i	$0.144933\pi$
$947$	27.0652i	0.879502i	−0.898120	+	0.439751i	$0.855067\pi$
$948$	0	0
$949$	−3.42047	−0.111033
$950$	0	0
$951$	−29.3096	−0.950430
$952$	0	0
$953$	− 9.21600i	− 0.298536i	−0.988797	−	0.149268i	$-0.952308\pi$
$953$	− 9.21600i	− 0.298536i	0.988797	−	0.149268i	$-0.0476917\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 32.7030i	− 1.05714i
$958$	0	0
$959$	42.0089	1.35654
$960$	0	0
$961$	−30.9246	−0.997569
$962$	0	0
$963$	− 3.74176i	− 0.120577i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 19.7979i	− 0.636658i	−0.947980	−	0.318329i	$-0.896878\pi$
$967$	− 19.7979i	− 0.636658i	0.947980	−	0.318329i	$-0.103122\pi$
$968$	0	0
$969$	7.60265	0.244232
$970$	0	0
$971$	−7.67033	−0.246153	−0.123076	−	0.992397i	$-0.539276\pi$
$971$	−7.67033	−0.246153	−0.123076	+	0.992397i	$0.539276\pi$
$972$	0	0
$973$	− 39.3626i	− 1.26191i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 44.5794i	− 1.42622i	−0.701052	−	0.713110i	$-0.747286\pi$
$977$	− 44.5794i	− 1.42622i	0.701052	−	0.713110i	$-0.252714\pi$
$978$	0	0
$979$	−27.5978	−0.882029
$980$	0	0
$981$	−97.6199	−3.11676
$982$	0	0
$983$	42.1176i	1.34334i	0.740849	+	0.671671i	$0.234423\pi$
$983$	42.1176i	1.34334i	−0.740849	+	0.671671i	$0.765577\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 47.6331i	− 1.51618i
$988$	0	0
$989$	−86.2833	−2.74365
$990$	0	0
$991$	20.2174	0.642228	0.321114	−	0.947040i	$-0.395943\pi$
$991$	20.2174	0.642228	0.321114	+	0.947040i	$0.395943\pi$
$992$	0	0
$993$	23.5734i	0.748079i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	47.9644i	1.51905i	0.650479	+	0.759524i	$0.274568\pi$
$997$	47.9644i	1.51905i	−0.650479	+	0.759524i	$0.725432\pi$
$998$	0	0
$999$	−4.54012	−0.143643

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4100.2.d.f.1149.1		14
5.2	odd	4		4100.2.a.h.1.1	✓	7
5.3	odd	4		4100.2.a.i.1.7	yes	7
5.4	even	2	inner	4100.2.d.f.1149.14		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4100.2.a.h.1.1	✓	7	5.2	odd	4
4100.2.a.i.1.7	yes	7	5.3	odd	4
4100.2.d.f.1149.1		14	1.1	even	1	trivial
4100.2.d.f.1149.14		14	5.4	even	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:	\( N \)	\(=\)	\( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4100.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.88961i q^{3} -4.40706i q^{7} -5.34984 q^{9} +O(q^{10})\)	\(q-2.88961i q^{3} -4.40706i q^{7} -5.34984 q^{9} +3.64440 q^{11} -0.559720i q^{13} +0.883584i q^{17} +2.97768 q^{19} -12.7347 q^{21} -9.05146i q^{23} +6.79012i q^{27} +3.10543 q^{29} -0.274514 q^{31} -10.5309i q^{33} +0.668636i q^{37} -1.61737 q^{39} -1.00000 q^{41} -9.53253i q^{43} +3.74043i q^{47} -12.4221 q^{49} +2.55321 q^{51} -9.11015i q^{53} -8.60433i q^{57} -10.5109 q^{59} -9.12149 q^{61} +23.5770i q^{63} +4.89074i q^{67} -26.1552 q^{69} +12.7884 q^{71} -6.11104i q^{73} -16.0611i q^{77} +11.8396 q^{79} +3.57127 q^{81} -0.381590i q^{83} -8.97349i q^{87} -7.57265 q^{89} -2.46672 q^{91} +0.793239i q^{93} +16.0284i q^{97} -19.4970 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 12 q^{9}+O(q^{10}) \) 14 * q - 12 * q^9	\( 14 q - 12 q^{9} - 6 q^{11} - 6 q^{19} - 26 q^{21} + 2 q^{29} - 8 q^{31} - 26 q^{39} - 14 q^{41} - 34 q^{49} - 42 q^{51} - 40 q^{59} - 20 q^{61} - 50 q^{69} - 46 q^{71} - 22 q^{79} - 26 q^{81} - 16 q^{89} - 38 q^{91} - 72 q^{99}+O(q^{100}) \) 14 * q - 12 * q^9 - 6 * q^11 - 6 * q^19 - 26 * q^21 + 2 * q^29 - 8 * q^31 - 26 * q^39 - 14 * q^41 - 34 * q^49 - 42 * q^51 - 40 * q^59 - 20 * q^61 - 50 * q^69 - 46 * q^71 - 22 * q^79 - 26 * q^81 - 16 * q^89 - 38 * q^91 - 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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