LMFDB - Embedded newform 4000.2.c.c.1249.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,2,Mod(1249,4000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4000, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4000.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4000 = 2^{5} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4000.c (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:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.6
Root			$-0.951057 + 0.309017i$ of defining polynomial
Character	$\chi$	$=$	4000.1249
Dual form			4000.2.c.c.1249.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.17557i q^{3} -0.726543i q^{7} +1.61803 q^{9} +O(q^{10})$	$q+1.17557i q^{3} -0.726543i q^{7} +1.61803 q^{9} +3.07768 q^{11} -0.381966i q^{13} +0.236068i q^{17} +4.97980 q^{19} +0.854102 q^{21} -7.60845i q^{23} +5.42882i q^{27} +5.00000 q^{29} -6.60440 q^{31} +3.61803i q^{33} -2.47214i q^{37} +0.449028 q^{39} -0.527864 q^{41} -6.88191i q^{43} +0.449028i q^{47} +6.47214 q^{49} -0.277515 q^{51} -6.32624i q^{53} +5.85410i q^{57} -11.3067 q^{59} +0.854102 q^{61} -1.17557i q^{63} -10.9637i q^{67} +8.94427 q^{69} +3.07768 q^{71} +2.61803i q^{73} -2.23607i q^{77} -5.98385 q^{79} -1.52786 q^{81} +5.87785i q^{87} +14.4721 q^{89} -0.277515 q^{91} -7.76393i q^{93} -3.85410i q^{97} +4.97980 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{9}+O(q^{10}) $ 8 * q + 4 * q^9	$ 8 q + 4 q^{9} - 20 q^{21} + 40 q^{29} - 40 q^{41} + 16 q^{49} - 20 q^{61} - 48 q^{81} + 80 q^{89}+O(q^{100}) $ 8 * q + 4 * q^9 - 20 * q^21 + 40 * q^29 - 40 * q^41 + 16 * q^49 - 20 * q^61 - 48 * q^81 + 80 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$.

$n$	$1377$	$2501$	$2751$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	1.17557i	0.678716i	0.940657	+	0.339358i	$0.110210\pi$
$3$	1.17557i	0.678716i	−0.940657	+	0.339358i	$0.889790\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.726543i	− 0.274607i	−0.990529	−	0.137304i	$-0.956156\pi$
$7$	− 0.726543i	− 0.274607i	0.990529	−	0.137304i	$-0.0438436\pi$
$8$	0	0
$9$	1.61803	0.539345
$10$	0	0
$11$	3.07768	0.927957	0.463978	−	0.885847i	$-0.346422\pi$
$11$	3.07768	0.927957	0.463978	+	0.885847i	$0.346422\pi$
$12$	0	0
$13$	− 0.381966i	− 0.105938i	−0.998596	−	0.0529692i	$-0.983131\pi$
$13$	− 0.381966i	− 0.105938i	0.998596	−	0.0529692i	$-0.0168685\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.236068i	0.0572549i	0.999590	+	0.0286274i	$0.00911364\pi$
$17$	0.236068i	0.0572549i	−0.999590	+	0.0286274i	$0.990886\pi$
$18$	0	0
$19$	4.97980	1.14244	0.571222	−	0.820796i	$-0.306470\pi$
$19$	4.97980	1.14244	0.571222	+	0.820796i	$0.306470\pi$
$20$	0	0
$21$	0.854102	0.186380
$22$	0	0
$23$	− 7.60845i	− 1.58647i	−0.608914	−	0.793236i	$-0.708395\pi$
$23$	− 7.60845i	− 1.58647i	0.608914	−	0.793236i	$-0.291605\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.42882i	1.04478i
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	−6.60440	−1.18618	−0.593092	−	0.805135i	$-0.702093\pi$
$31$	−6.60440	−1.18618	−0.593092	+	0.805135i	$0.702093\pi$
$32$	0	0
$33$	3.61803i	0.629819i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.47214i	− 0.406417i	−0.979136	−	0.203208i	$-0.934863\pi$
$37$	− 2.47214i	− 0.406417i	0.979136	−	0.203208i	$-0.0651369\pi$
$38$	0	0
$39$	0.449028	0.0719020
$40$	0	0
$41$	−0.527864	−0.0824385	−0.0412193	−	0.999150i	$-0.513124\pi$
$41$	−0.527864	−0.0824385	−0.0412193	+	0.999150i	$0.513124\pi$
$42$	0	0
$43$	− 6.88191i	− 1.04948i	−0.851262	−	0.524741i	$-0.824162\pi$
$43$	− 6.88191i	− 1.04948i	0.851262	−	0.524741i	$-0.175838\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.449028i	0.0654975i	0.999464	+	0.0327487i	$0.0104261\pi$
$47$	0.449028i	0.0654975i	−0.999464	+	0.0327487i	$0.989574\pi$
$48$	0	0
$49$	6.47214	0.924591
$50$	0	0
$51$	−0.277515	−0.0388598
$52$	0	0
$53$	− 6.32624i	− 0.868976i	−0.900678	−	0.434488i	$-0.856929\pi$
$53$	− 6.32624i	− 0.868976i	0.900678	−	0.434488i	$-0.143071\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.85410i	0.775395i
$58$	0	0
$59$	−11.3067	−1.47200	−0.736002	−	0.676979i	$-0.763289\pi$
$59$	−11.3067	−1.47200	−0.736002	+	0.676979i	$0.763289\pi$
$60$	0	0
$61$	0.854102	0.109357	0.0546783	−	0.998504i	$-0.482587\pi$
$61$	0.854102	0.109357	0.0546783	+	0.998504i	$0.482587\pi$
$62$	0	0
$63$	− 1.17557i	− 0.148108i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 10.9637i	− 1.33942i	−0.742621	−	0.669712i	$-0.766418\pi$
$67$	− 10.9637i	− 1.33942i	0.742621	−	0.669712i	$-0.233582\pi$
$68$	0	0
$69$	8.94427	1.07676
$70$	0	0
$71$	3.07768	0.365254	0.182627	−	0.983182i	$-0.441540\pi$
$71$	3.07768	0.365254	0.182627	+	0.983182i	$0.441540\pi$
$72$	0	0
$73$	2.61803i	0.306418i	0.988194	+	0.153209i	$0.0489607\pi$
$73$	2.61803i	0.306418i	−0.988194	+	0.153209i	$0.951039\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.23607i	− 0.254824i
$78$	0	0
$79$	−5.98385	−0.673236	−0.336618	−	0.941641i	$-0.609283\pi$
$79$	−5.98385	−0.673236	−0.336618	+	0.941641i	$0.609283\pi$
$80$	0	0
$81$	−1.52786	−0.169763
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.87785i	0.630172i
$88$	0	0
$89$	14.4721	1.53404	0.767022	−	0.641621i	$-0.221738\pi$
$89$	14.4721	1.53404	0.767022	+	0.641621i	$0.221738\pi$
$90$	0	0
$91$	−0.277515	−0.0290914
$92$	0	0
$93$	− 7.76393i	− 0.805082i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 3.85410i	− 0.391325i	−0.980671	−	0.195662i	$-0.937314\pi$
$97$	− 3.85410i	− 0.391325i	0.980671	−	0.195662i	$-0.0626857\pi$
$98$	0	0
$99$	4.97980	0.500488
$100$	0	0
$101$	−7.00000	−0.696526	−0.348263	−	0.937397i	$-0.613228\pi$
$101$	−7.00000	−0.696526	−0.348263	+	0.937397i	$0.613228\pi$
$102$	0	0
$103$	2.17963i	0.214765i	0.994218	+	0.107383i	$0.0342470\pi$
$103$	2.17963i	0.214765i	−0.994218	+	0.107383i	$0.965753\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.17557i	− 0.113647i	−0.998384	−	0.0568233i	$-0.981903\pi$
$107$	− 1.17557i	− 0.113647i	0.998384	−	0.0568233i	$-0.0180972\pi$
$108$	0	0
$109$	12.0902	1.15803	0.579014	−	0.815318i	$-0.303438\pi$
$109$	12.0902	1.15803	0.579014	+	0.815318i	$0.303438\pi$
$110$	0	0
$111$	2.90617	0.275841
$112$	0	0
$113$	17.1803i	1.61619i	0.589052	+	0.808095i	$0.299501\pi$
$113$	17.1803i	1.61619i	−0.589052	+	0.808095i	$0.700499\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 0.618034i	− 0.0571373i
$118$	0	0
$119$	0.171513	0.0157226
$120$	0	0
$121$	−1.52786	−0.138897
$122$	0	0
$123$	− 0.620541i	− 0.0559523i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 7.60845i	− 0.675141i	−0.941300	−	0.337570i	$-0.890395\pi$
$127$	− 7.60845i	− 0.675141i	0.941300	−	0.337570i	$-0.109605\pi$
$128$	0	0
$129$	8.09017	0.712300
$130$	0	0
$131$	15.4944	1.35375	0.676877	−	0.736096i	$-0.263333\pi$
$131$	15.4944	1.35375	0.676877	+	0.736096i	$0.263333\pi$
$132$	0	0
$133$	− 3.61803i	− 0.313723i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.67376i	− 0.228435i	−0.993456	−	0.114217i	$-0.963564\pi$
$137$	− 2.67376i	− 0.228435i	0.993456	−	0.114217i	$-0.0364361\pi$
$138$	0	0
$139$	7.43694	0.630793	0.315396	−	0.948960i	$-0.397863\pi$
$139$	7.43694	0.630793	0.315396	+	0.948960i	$0.397863\pi$
$140$	0	0
$141$	−0.527864	−0.0444542
$142$	0	0
$143$	− 1.17557i	− 0.0983061i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	7.60845i	0.627535i
$148$	0	0
$149$	15.4721	1.26753	0.633763	−	0.773527i	$-0.281509\pi$
$149$	15.4721	1.26753	0.633763	+	0.773527i	$0.281509\pi$
$150$	0	0
$151$	−16.2210	−1.32004	−0.660022	−	0.751247i	$-0.729453\pi$
$151$	−16.2210	−1.32004	−0.660022	+	0.751247i	$0.729453\pi$
$152$	0	0
$153$	0.381966i	0.0308801i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.8541i	1.50472i	0.658752	+	0.752361i	$0.271085\pi$
$157$	18.8541i	1.50472i	−0.658752	+	0.752361i	$0.728915\pi$
$158$	0	0
$159$	7.43694	0.589788
$160$	0	0
$161$	−5.52786	−0.435657
$162$	0	0
$163$	− 8.05748i	− 0.631111i	−0.948907	−	0.315555i	$-0.897809\pi$
$163$	− 8.05748i	− 0.631111i	0.948907	−	0.315555i	$-0.102191\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.81234i	0.449772i	0.974385	+	0.224886i	$0.0722010\pi$
$167$	5.81234i	0.449772i	−0.974385	+	0.224886i	$0.927799\pi$
$168$	0	0
$169$	12.8541	0.988777
$170$	0	0
$171$	8.05748	0.616171
$172$	0	0
$173$	− 21.6525i	− 1.64621i	−0.567891	−	0.823104i	$-0.692241\pi$
$173$	− 21.6525i	− 1.64621i	0.567891	−	0.823104i	$-0.307759\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 13.2918i	− 0.999073i
$178$	0	0
$179$	0.171513	0.0128195	0.00640976	−	0.999979i	$-0.497960\pi$
$179$	0.171513	0.0128195	0.00640976	+	0.999979i	$0.497960\pi$
$180$	0	0
$181$	9.88854	0.735010	0.367505	−	0.930022i	$-0.380212\pi$
$181$	9.88854	0.735010	0.367505	+	0.930022i	$0.380212\pi$
$182$	0	0
$183$	1.00406i	0.0742220i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.726543i	0.0531301i
$188$	0	0
$189$	3.94427	0.286904
$190$	0	0
$191$	−22.8254	−1.65158	−0.825792	−	0.563974i	$-0.809272\pi$
$191$	−22.8254	−1.65158	−0.825792	+	0.563974i	$0.809272\pi$
$192$	0	0
$193$	11.8541i	0.853277i	0.904422	+	0.426638i	$0.140302\pi$
$193$	11.8541i	0.853277i	−0.904422	+	0.426638i	$0.859698\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.65248i	− 0.260228i	−0.991499	−	0.130114i	$-0.958466\pi$
$197$	− 3.65248i	− 0.260228i	0.991499	−	0.130114i	$-0.0415344\pi$
$198$	0	0
$199$	−5.81234	−0.412026	−0.206013	−	0.978549i	$-0.566049\pi$
$199$	−5.81234	−0.412026	−0.206013	+	0.978549i	$0.566049\pi$
$200$	0	0
$201$	12.8885	0.909088
$202$	0	0
$203$	− 3.63271i	− 0.254966i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 12.3107i	− 0.855655i
$208$	0	0
$209$	15.3262	1.06014
$210$	0	0
$211$	−1.79611	−0.123649	−0.0618247	−	0.998087i	$-0.519692\pi$
$211$	−1.79611	−0.123649	−0.0618247	+	0.998087i	$0.519692\pi$
$212$	0	0
$213$	3.61803i	0.247904i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.79837i	0.325735i
$218$	0	0
$219$	−3.07768	−0.207971
$220$	0	0
$221$	0.0901699	0.00606549
$222$	0	0
$223$	22.1643i	1.48423i	0.670271	+	0.742117i	$0.266178\pi$
$223$	22.1643i	1.48423i	−0.670271	+	0.742117i	$0.733822\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 9.85359i	− 0.654006i	−0.945023	−	0.327003i	$-0.893961\pi$
$227$	− 9.85359i	− 0.654006i	0.945023	−	0.327003i	$-0.106039\pi$
$228$	0	0
$229$	17.5623	1.16055	0.580275	−	0.814421i	$-0.302945\pi$
$229$	17.5623	1.16055	0.580275	+	0.814421i	$0.302945\pi$
$230$	0	0
$231$	2.62866	0.172953
$232$	0	0
$233$	21.6525i	1.41850i	0.704957	+	0.709250i	$0.250966\pi$
$233$	21.6525i	1.41850i	−0.704957	+	0.709250i	$0.749034\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 7.03444i	− 0.456936i
$238$	0	0
$239$	20.9232	1.35341	0.676706	−	0.736253i	$-0.263407\pi$
$239$	20.9232	1.35341	0.676706	+	0.736253i	$0.263407\pi$
$240$	0	0
$241$	9.47214	0.610154	0.305077	−	0.952328i	$-0.401318\pi$
$241$	9.47214	0.610154	0.305077	+	0.952328i	$0.401318\pi$
$242$	0	0
$243$	14.4904i	0.929557i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.90211i	− 0.121029i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−7.88597	−0.497758	−0.248879	−	0.968535i	$-0.580062\pi$
$251$	−7.88597	−0.497758	−0.248879	+	0.968535i	$0.580062\pi$
$252$	0	0
$253$	− 23.4164i	− 1.47218i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.2361i	0.638508i	0.947669	+	0.319254i	$0.103432\pi$
$257$	10.2361i	0.638508i	−0.947669	+	0.319254i	$0.896568\pi$
$258$	0	0
$259$	−1.79611	−0.111605
$260$	0	0
$261$	8.09017	0.500769
$262$	0	0
$263$	21.2663i	1.31133i	0.755050	+	0.655667i	$0.227613\pi$
$263$	21.2663i	1.31133i	−0.755050	+	0.655667i	$0.772387\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	17.0130i	1.04118i
$268$	0	0
$269$	7.90983	0.482271	0.241135	−	0.970491i	$-0.422480\pi$
$269$	7.90983	0.482271	0.241135	+	0.970491i	$0.422480\pi$
$270$	0	0
$271$	25.4540	1.54622	0.773111	−	0.634271i	$-0.218700\pi$
$271$	25.4540	1.54622	0.773111	+	0.634271i	$0.218700\pi$
$272$	0	0
$273$	− 0.326238i	− 0.0197448i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 14.2361i	− 0.855362i	−0.903930	−	0.427681i	$-0.859331\pi$
$277$	− 14.2361i	− 0.855362i	0.903930	−	0.427681i	$-0.140669\pi$
$278$	0	0
$279$	−10.6861	−0.639762
$280$	0	0
$281$	−17.8885	−1.06714	−0.533571	−	0.845756i	$-0.679150\pi$
$281$	−17.8885	−1.06714	−0.533571	+	0.845756i	$0.679150\pi$
$282$	0	0
$283$	26.4176i	1.57036i	0.619266	+	0.785181i	$0.287430\pi$
$283$	26.4176i	1.57036i	−0.619266	+	0.785181i	$0.712570\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.383516i	0.0226382i
$288$	0	0
$289$	16.9443	0.996722
$290$	0	0
$291$	4.53077	0.265598
$292$	0	0
$293$	− 10.7082i	− 0.625580i	−0.949822	−	0.312790i	$-0.898736\pi$
$293$	− 10.7082i	− 0.625580i	0.949822	−	0.312790i	$-0.101264\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	16.7082i	0.969508i
$298$	0	0
$299$	−2.90617	−0.168068
$300$	0	0
$301$	−5.00000	−0.288195
$302$	0	0
$303$	− 8.22899i	− 0.472743i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	24.0009i	1.36981i	0.728635	+	0.684903i	$0.240155\pi$
$307$	24.0009i	1.36981i	−0.728635	+	0.684903i	$0.759845\pi$
$308$	0	0
$309$	−2.56231	−0.145764
$310$	0	0
$311$	16.0090	0.907785	0.453892	−	0.891057i	$-0.350035\pi$
$311$	16.0090	0.907785	0.453892	+	0.891057i	$0.350035\pi$
$312$	0	0
$313$	22.8328i	1.29059i	0.763935	+	0.645294i	$0.223265\pi$
$313$	22.8328i	1.29059i	−0.763935	+	0.645294i	$0.776735\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.9443i	0.951685i	0.879530	+	0.475843i	$0.157857\pi$
$317$	16.9443i	0.951685i	−0.879530	+	0.475843i	$0.842143\pi$
$318$	0	0
$319$	15.3884	0.861586
$320$	0	0
$321$	1.38197	0.0771338
$322$	0	0
$323$	1.17557i	0.0654105i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	14.2128i	0.785972i
$328$	0	0
$329$	0.326238	0.0179861
$330$	0	0
$331$	−18.0171	−0.990308	−0.495154	−	0.868805i	$-0.664888\pi$
$331$	−18.0171	−0.990308	−0.495154	+	0.868805i	$0.664888\pi$
$332$	0	0
$333$	− 4.00000i	− 0.219199i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 33.6525i	− 1.83317i	−0.399843	−	0.916584i	$-0.630935\pi$
$337$	− 33.6525i	− 1.83317i	0.399843	−	0.916584i	$-0.369065\pi$
$338$	0	0
$339$	−20.1967	−1.09693
$340$	0	0
$341$	−20.3262	−1.10073
$342$	0	0
$343$	− 9.78808i	− 0.528507i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 31.8464i	− 1.70960i	−0.518954	−	0.854802i	$-0.673678\pi$
$347$	− 31.8464i	− 1.70960i	0.518954	−	0.854802i	$-0.326322\pi$
$348$	0	0
$349$	12.8885	0.689908	0.344954	−	0.938620i	$-0.387895\pi$
$349$	12.8885	0.689908	0.344954	+	0.938620i	$0.387895\pi$
$350$	0	0
$351$	2.07363	0.110682
$352$	0	0
$353$	0.583592i	0.0310615i	0.999879	+	0.0155307i	$0.00494379\pi$
$353$	0.583592i	0.0310615i	−0.999879	+	0.0155307i	$0.995056\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.201626i	0.0106712i
$358$	0	0
$359$	−31.9524	−1.68638	−0.843192	−	0.537613i	$-0.819326\pi$
$359$	−31.9524	−1.68638	−0.843192	+	0.537613i	$0.819326\pi$
$360$	0	0
$361$	5.79837	0.305178
$362$	0	0
$363$	− 1.79611i	− 0.0942714i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	14.0413i	0.732952i	0.930428	+	0.366476i	$0.119436\pi$
$367$	14.0413i	0.732952i	−0.930428	+	0.366476i	$0.880564\pi$
$368$	0	0
$369$	−0.854102	−0.0444628
$370$	0	0
$371$	−4.59628	−0.238627
$372$	0	0
$373$	− 35.2705i	− 1.82624i	−0.407693	−	0.913119i	$-0.633667\pi$
$373$	− 35.2705i	− 1.82624i	0.407693	−	0.913119i	$-0.366333\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1.90983i	− 0.0983613i
$378$	0	0
$379$	−18.9151	−0.971605	−0.485802	−	0.874069i	$-0.661473\pi$
$379$	−18.9151	−0.971605	−0.485802	+	0.874069i	$0.661473\pi$
$380$	0	0
$381$	8.94427	0.458229
$382$	0	0
$383$	− 31.5034i	− 1.60975i	−0.593446	−	0.804874i	$-0.702233\pi$
$383$	− 31.5034i	− 1.60975i	0.593446	−	0.804874i	$-0.297767\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 11.1352i	− 0.566032i
$388$	0	0
$389$	−14.4164	−0.730941	−0.365470	−	0.930823i	$-0.619092\pi$
$389$	−14.4164	−0.730941	−0.365470	+	0.930823i	$0.619092\pi$
$390$	0	0
$391$	1.79611	0.0908333
$392$	0	0
$393$	18.2148i	0.918814i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	19.7639i	0.991923i	0.868344	+	0.495962i	$0.165184\pi$
$397$	19.7639i	0.991923i	−0.868344	+	0.495962i	$0.834816\pi$
$398$	0	0
$399$	4.25325	0.212929
$400$	0	0
$401$	−4.58359	−0.228894	−0.114447	−	0.993429i	$-0.536510\pi$
$401$	−4.58359	−0.228894	−0.114447	+	0.993429i	$0.536510\pi$
$402$	0	0
$403$	2.52265i	0.125662i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 7.60845i	− 0.377137i
$408$	0	0
$409$	−0.583592	−0.0288568	−0.0144284	−	0.999896i	$-0.504593\pi$
$409$	−0.583592	−0.0288568	−0.0144284	+	0.999896i	$0.504593\pi$
$410$	0	0
$411$	3.14320	0.155042
$412$	0	0
$413$	8.21478i	0.404223i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.74265i	0.428129i
$418$	0	0
$419$	38.5568	1.88362	0.941811	−	0.336142i	$-0.109122\pi$
$419$	38.5568	1.88362	0.941811	+	0.336142i	$0.109122\pi$
$420$	0	0
$421$	8.61803	0.420017	0.210009	−	0.977700i	$-0.432651\pi$
$421$	8.61803	0.420017	0.210009	+	0.977700i	$0.432651\pi$
$422$	0	0
$423$	0.726543i	0.0353257i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 0.620541i	− 0.0300301i
$428$	0	0
$429$	1.38197	0.0667219
$430$	0	0
$431$	−1.51860	−0.0731483	−0.0365741	−	0.999331i	$-0.511645\pi$
$431$	−1.51860	−0.0731483	−0.0365741	+	0.999331i	$0.511645\pi$
$432$	0	0
$433$	29.5279i	1.41902i	0.704696	+	0.709509i	$0.251083\pi$
$433$	29.5279i	1.41902i	−0.704696	+	0.709509i	$0.748917\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 37.8885i	− 1.81245i
$438$	0	0
$439$	−17.1845	−0.820173	−0.410086	−	0.912047i	$-0.634501\pi$
$439$	−17.1845	−0.820173	−0.410086	+	0.912047i	$0.634501\pi$
$440$	0	0
$441$	10.4721	0.498673
$442$	0	0
$443$	30.0503i	1.42773i	0.700282	+	0.713866i	$0.253058\pi$
$443$	30.0503i	1.42773i	−0.700282	+	0.713866i	$0.746942\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	18.1886i	0.860291i
$448$	0	0
$449$	−16.9787	−0.801275	−0.400638	−	0.916237i	$-0.631211\pi$
$449$	−16.9787	−0.801275	−0.400638	+	0.916237i	$0.631211\pi$
$450$	0	0
$451$	−1.62460	−0.0764994
$452$	0	0
$453$	− 19.0689i	− 0.895934i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	29.3050i	1.37083i	0.728154	+	0.685414i	$0.240379\pi$
$457$	29.3050i	1.37083i	−0.728154	+	0.685414i	$0.759621\pi$
$458$	0	0
$459$	−1.28157	−0.0598186
$460$	0	0
$461$	−16.2705	−0.757793	−0.378897	−	0.925439i	$-0.623696\pi$
$461$	−16.2705	−0.757793	−0.378897	+	0.925439i	$0.623696\pi$
$462$	0	0
$463$	3.97574i	0.184768i	0.995723	+	0.0923841i	$0.0294488\pi$
$463$	3.97574i	0.184768i	−0.995723	+	0.0923841i	$0.970551\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 18.1231i	− 0.838636i	−0.907839	−	0.419318i	$-0.862269\pi$
$467$	− 18.1231i	− 0.838636i	0.907839	−	0.419318i	$-0.137731\pi$
$468$	0	0
$469$	−7.96556	−0.367815
$470$	0	0
$471$	−22.1643	−1.02128
$472$	0	0
$473$	− 21.1803i	− 0.973873i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 10.2361i	− 0.468677i
$478$	0	0
$479$	8.44100	0.385679	0.192839	−	0.981230i	$-0.438230\pi$
$479$	8.44100	0.385679	0.192839	+	0.981230i	$0.438230\pi$
$480$	0	0
$481$	−0.944272	−0.0430551
$482$	0	0
$483$	− 6.49839i	− 0.295687i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 35.1361i	− 1.59217i	−0.605186	−	0.796084i	$-0.706901\pi$
$487$	− 35.1361i	− 1.59217i	0.605186	−	0.796084i	$-0.293099\pi$
$488$	0	0
$489$	9.47214	0.428345
$490$	0	0
$491$	−17.8456	−0.805359	−0.402679	−	0.915341i	$-0.631921\pi$
$491$	−17.8456	−0.805359	−0.402679	+	0.915341i	$0.631921\pi$
$492$	0	0
$493$	1.18034i	0.0531598i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2.23607i	− 0.100301i
$498$	0	0
$499$	23.5114	1.05252	0.526258	−	0.850325i	$-0.323595\pi$
$499$	23.5114	1.05252	0.526258	+	0.850325i	$0.323595\pi$
$500$	0	0
$501$	−6.83282	−0.305268
$502$	0	0
$503$	24.0009i	1.07015i	0.844805	+	0.535074i	$0.179716\pi$
$503$	24.0009i	1.07015i	−0.844805	+	0.535074i	$0.820284\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	15.1109i	0.671099i
$508$	0	0
$509$	−8.94427	−0.396448	−0.198224	−	0.980157i	$-0.563517\pi$
$509$	−8.94427	−0.396448	−0.198224	+	0.980157i	$0.563517\pi$
$510$	0	0
$511$	1.90211	0.0841445
$512$	0	0
$513$	27.0344i	1.19360i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.38197i	0.0607788i
$518$	0	0
$519$	25.4540	1.11731
$520$	0	0
$521$	−39.6869	−1.73872	−0.869358	−	0.494183i	$-0.835467\pi$
$521$	−39.6869	−1.73872	−0.869358	+	0.494183i	$0.835467\pi$
$522$	0	0
$523$	− 15.9030i	− 0.695388i	−0.937608	−	0.347694i	$-0.886965\pi$
$523$	− 15.9030i	− 0.695388i	0.937608	−	0.347694i	$-0.113035\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.55909i	− 0.0679149i
$528$	0	0
$529$	−34.8885	−1.51689
$530$	0	0
$531$	−18.2946	−0.793917
$532$	0	0
$533$	0.201626i	0.00873340i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.201626i	0.00870081i
$538$	0	0
$539$	19.9192	0.857980
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	0	0
$543$	11.6247i	0.498863i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14.5559i	0.622364i	0.950350	+	0.311182i	$0.100725\pi$
$547$	14.5559i	0.622364i	−0.950350	+	0.311182i	$0.899275\pi$
$548$	0	0
$549$	1.38197	0.0589809
$550$	0	0
$551$	24.8990	1.06073
$552$	0	0
$553$	4.34752i	0.184876i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	26.4721i	1.12166i	0.827931	+	0.560830i	$0.189518\pi$
$557$	26.4721i	1.12166i	−0.827931	+	0.560830i	$0.810482\pi$
$558$	0	0
$559$	−2.62866	−0.111180
$560$	0	0
$561$	−0.854102	−0.0360602
$562$	0	0
$563$	− 27.5276i	− 1.16015i	−0.814562	−	0.580076i	$-0.803023\pi$
$563$	− 27.5276i	− 1.16015i	0.814562	−	0.580076i	$-0.196977\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.11006i	0.0466181i
$568$	0	0
$569$	−8.36068	−0.350498	−0.175249	−	0.984524i	$-0.556073\pi$
$569$	−8.36068	−0.350498	−0.175249	+	0.984524i	$0.556073\pi$
$570$	0	0
$571$	32.0584	1.34160	0.670801	−	0.741637i	$-0.265950\pi$
$571$	32.0584	1.34160	0.670801	+	0.741637i	$0.265950\pi$
$572$	0	0
$573$	− 26.8328i	− 1.12096i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 33.3820i	− 1.38971i	−0.719150	−	0.694855i	$-0.755469\pi$
$577$	− 33.3820i	− 1.38971i	0.719150	−	0.694855i	$-0.244531\pi$
$578$	0	0
$579$	−13.9353	−0.579133
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 19.4702i	− 0.806372i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11.2007i	0.462301i	0.972918	+	0.231151i	$0.0742490\pi$
$587$	11.2007i	0.462301i	−0.972918	+	0.231151i	$0.925751\pi$
$588$	0	0
$589$	−32.8885	−1.35515
$590$	0	0
$591$	4.29374	0.176621
$592$	0	0
$593$	8.59675i	0.353026i	0.984298	+	0.176513i	$0.0564818\pi$
$593$	8.59675i	0.353026i	−0.984298	+	0.176513i	$0.943518\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 6.83282i	− 0.279649i
$598$	0	0
$599$	5.53483	0.226147	0.113073	−	0.993587i	$-0.463930\pi$
$599$	5.53483	0.226147	0.113073	+	0.993587i	$0.463930\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	0	0
$603$	− 17.7396i	− 0.722411i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 2.90617i	− 0.117958i	−0.998259	−	0.0589789i	$-0.981216\pi$
$607$	− 2.90617i	− 0.117958i	0.998259	−	0.0589789i	$-0.0187845\pi$
$608$	0	0
$609$	4.27051	0.173050
$610$	0	0
$611$	0.171513	0.00693869
$612$	0	0
$613$	7.18034i	0.290011i	0.989431	+	0.145006i	$0.0463200\pi$
$613$	7.18034i	0.290011i	−0.989431	+	0.145006i	$0.953680\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	37.0689i	1.49234i	0.665757	+	0.746169i	$0.268109\pi$
$617$	37.0689i	1.49234i	−0.665757	+	0.746169i	$0.731891\pi$
$618$	0	0
$619$	−33.5115	−1.34694	−0.673470	−	0.739214i	$-0.735197\pi$
$619$	−33.5115	−1.34694	−0.673470	+	0.739214i	$0.735197\pi$
$620$	0	0
$621$	41.3050	1.65751
$622$	0	0
$623$	− 10.5146i	− 0.421259i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	18.0171i	0.719533i
$628$	0	0
$629$	0.583592	0.0232693
$630$	0	0
$631$	35.8221	1.42606	0.713029	−	0.701135i	$-0.247323\pi$
$631$	35.8221	1.42606	0.713029	+	0.701135i	$0.247323\pi$
$632$	0	0
$633$	− 2.11146i	− 0.0839228i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 2.47214i	− 0.0979496i
$638$	0	0
$639$	4.97980	0.196998
$640$	0	0
$641$	−0.326238	−0.0128856	−0.00644281	−	0.999979i	$-0.502051\pi$
$641$	−0.326238	−0.0128856	−0.00644281	+	0.999979i	$0.502051\pi$
$642$	0	0
$643$	− 19.2986i	− 0.761064i	−0.924768	−	0.380532i	$-0.875741\pi$
$643$	− 19.2986i	− 0.761064i	0.924768	−	0.380532i	$-0.124259\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 5.08580i	− 0.199943i	−0.994990	−	0.0999717i	$-0.968125\pi$
$647$	− 5.08580i	− 0.199943i	0.994990	−	0.0999717i	$-0.0318752\pi$
$648$	0	0
$649$	−34.7984	−1.36596
$650$	0	0
$651$	−5.64083	−0.221081
$652$	0	0
$653$	− 16.0000i	− 0.626128i	−0.949732	−	0.313064i	$-0.898644\pi$
$653$	− 16.0000i	− 0.626128i	0.949732	−	0.313064i	$-0.101356\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.23607i	0.165265i
$658$	0	0
$659$	16.9070	0.658604	0.329302	−	0.944225i	$-0.393187\pi$
$659$	16.9070	0.658604	0.329302	+	0.944225i	$0.393187\pi$
$660$	0	0
$661$	11.0557	0.430018	0.215009	−	0.976612i	$-0.431022\pi$
$661$	11.0557	0.430018	0.215009	+	0.976612i	$0.431022\pi$
$662$	0	0
$663$	0.106001i	0.00411674i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 38.0423i	− 1.47300i
$668$	0	0
$669$	−26.0557	−1.00737
$670$	0	0
$671$	2.62866	0.101478
$672$	0	0
$673$	− 33.6869i	− 1.29854i	−0.760560	−	0.649268i	$-0.775076\pi$
$673$	− 33.6869i	− 1.29854i	0.760560	−	0.649268i	$-0.224924\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 45.8885i	− 1.76364i	−0.471586	−	0.881820i	$-0.656318\pi$
$677$	− 45.8885i	− 1.76364i	0.471586	−	0.881820i	$-0.343682\pi$
$678$	0	0
$679$	−2.80017	−0.107461
$680$	0	0
$681$	11.5836	0.443884
$682$	0	0
$683$	− 31.1604i	− 1.19232i	−0.802867	−	0.596159i	$-0.796693\pi$
$683$	− 31.1604i	− 1.19232i	0.802867	−	0.596159i	$-0.203307\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	20.6457i	0.787684i
$688$	0	0
$689$	−2.41641	−0.0920578
$690$	0	0
$691$	−35.1361	−1.33664	−0.668320	−	0.743874i	$-0.732986\pi$
$691$	−35.1361	−1.33664	−0.668320	+	0.743874i	$0.732986\pi$
$692$	0	0
$693$	− 3.61803i	− 0.137438i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 0.124612i	− 0.00472001i
$698$	0	0
$699$	−25.4540	−0.962759
$700$	0	0
$701$	−47.1591	−1.78117	−0.890586	−	0.454814i	$-0.849706\pi$
$701$	−47.1591	−1.78117	−0.890586	+	0.454814i	$0.849706\pi$
$702$	0	0
$703$	− 12.3107i	− 0.464308i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.08580i	0.191271i
$708$	0	0
$709$	−45.9787	−1.72677	−0.863383	−	0.504548i	$-0.831659\pi$
$709$	−45.9787	−1.72677	−0.863383	+	0.504548i	$0.831659\pi$
$710$	0	0
$711$	−9.68208	−0.363106
$712$	0	0
$713$	50.2492i	1.88185i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	24.5967i	0.918582i
$718$	0	0
$719$	−46.4428	−1.73202	−0.866011	−	0.500024i	$-0.833324\pi$
$719$	−46.4428	−1.73202	−0.866011	+	0.500024i	$0.833324\pi$
$720$	0	0
$721$	1.58359	0.0589761
$722$	0	0
$723$	11.1352i	0.414121i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 38.0423i	− 1.41091i	−0.708755	−	0.705455i	$-0.750743\pi$
$727$	− 38.0423i	− 1.41091i	0.708755	−	0.705455i	$-0.249257\pi$
$728$	0	0
$729$	−21.6180	−0.800668
$730$	0	0
$731$	1.62460	0.0600879
$732$	0	0
$733$	26.6869i	0.985704i	0.870113	+	0.492852i	$0.164046\pi$
$733$	26.6869i	0.985704i	−0.870113	+	0.492852i	$0.835954\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 33.7426i	− 1.24293i
$738$	0	0
$739$	20.8577	0.767264	0.383632	−	0.923486i	$-0.374673\pi$
$739$	20.8577	0.767264	0.383632	+	0.923486i	$0.374673\pi$
$740$	0	0
$741$	2.23607	0.0821440
$742$	0	0
$743$	− 33.9605i	− 1.24589i	−0.782265	−	0.622945i	$-0.785936\pi$
$743$	− 33.9605i	− 1.24589i	0.782265	−	0.622945i	$-0.214064\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−0.854102	−0.0312082
$750$	0	0
$751$	0.938545	0.0342480	0.0171240	−	0.999853i	$-0.494549\pi$
$751$	0.938545	0.0342480	0.0171240	+	0.999853i	$0.494549\pi$
$752$	0	0
$753$	− 9.27051i	− 0.337836i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	39.1803i	1.42403i	0.702162	+	0.712017i	$0.252218\pi$
$757$	39.1803i	1.42403i	−0.702162	+	0.712017i	$0.747782\pi$
$758$	0	0
$759$	27.5276	0.999190
$760$	0	0
$761$	−2.47214	−0.0896149	−0.0448074	−	0.998996i	$-0.514267\pi$
$761$	−2.47214	−0.0896149	−0.0448074	+	0.998996i	$0.514267\pi$
$762$	0	0
$763$	− 8.78402i	− 0.318003i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.31877i	0.155942i
$768$	0	0
$769$	−13.9443	−0.502843	−0.251422	−	0.967878i	$-0.580898\pi$
$769$	−13.9443	−0.502843	−0.251422	+	0.967878i	$0.580898\pi$
$770$	0	0
$771$	−12.0332	−0.433366
$772$	0	0
$773$	− 25.3050i	− 0.910156i	−0.890452	−	0.455078i	$-0.849611\pi$
$773$	− 25.3050i	− 0.910156i	0.890452	−	0.455078i	$-0.150389\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 2.11146i	− 0.0757481i
$778$	0	0
$779$	−2.62866	−0.0941814
$780$	0	0
$781$	9.47214	0.338940
$782$	0	0
$783$	27.1441i	0.970052i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 3.63271i	− 0.129492i	−0.997902	−	0.0647461i	$-0.979376\pi$
$787$	− 3.63271i	− 0.129492i	0.997902	−	0.0647461i	$-0.0206238\pi$
$788$	0	0
$789$	−25.0000	−0.890024
$790$	0	0
$791$	12.4822	0.443818
$792$	0	0
$793$	− 0.326238i	− 0.0115850i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24.3820i	0.863654i	0.901957	+	0.431827i	$0.142131\pi$
$797$	24.3820i	0.863654i	−0.901957	+	0.431827i	$0.857869\pi$
$798$	0	0
$799$	−0.106001	−0.00375005
$800$	0	0
$801$	23.4164	0.827378
$802$	0	0
$803$	8.05748i	0.284342i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	9.29856i	0.327325i
$808$	0	0
$809$	−30.2492	−1.06351	−0.531753	−	0.846899i	$-0.678467\pi$
$809$	−30.2492	−1.06351	−0.531753	+	0.846899i	$0.678467\pi$
$810$	0	0
$811$	−14.1068	−0.495358	−0.247679	−	0.968842i	$-0.579668\pi$
$811$	−14.1068	−0.495358	−0.247679	+	0.968842i	$0.579668\pi$
$812$	0	0
$813$	29.9230i	1.04944i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 34.2705i	− 1.19897i
$818$	0	0
$819$	−0.449028	−0.0156903
$820$	0	0
$821$	−3.09017	−0.107848	−0.0539238	−	0.998545i	$-0.517173\pi$
$821$	−3.09017	−0.107848	−0.0539238	+	0.998545i	$0.517173\pi$
$822$	0	0
$823$	− 17.0535i	− 0.594448i	−0.954808	−	0.297224i	$-0.903939\pi$
$823$	− 17.0535i	− 0.594448i	0.954808	−	0.297224i	$-0.0960608\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.4993i	1.06057i	0.847821	+	0.530283i	$0.177914\pi$
$827$	30.4993i	1.06057i	−0.847821	+	0.530283i	$0.822086\pi$
$828$	0	0
$829$	0.549150	0.0190728	0.00953639	−	0.999955i	$-0.496964\pi$
$829$	0.549150	0.0190728	0.00953639	+	0.999955i	$0.496964\pi$
$830$	0	0
$831$	16.7355	0.580548
$832$	0	0
$833$	1.52786i	0.0529374i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 35.8541i	− 1.23930i
$838$	0	0
$839$	−34.3035	−1.18429	−0.592145	−	0.805831i	$-0.701719\pi$
$839$	−34.3035	−1.18429	−0.592145	+	0.805831i	$0.701719\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	− 21.0292i	− 0.724286i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.11006i	0.0381421i
$848$	0	0
$849$	−31.0557	−1.06583
$850$	0	0
$851$	−18.8091	−0.644769
$852$	0	0
$853$	− 27.5410i	− 0.942987i	−0.881870	−	0.471493i	$-0.843715\pi$
$853$	− 27.5410i	− 0.942987i	0.881870	−	0.471493i	$-0.156285\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	46.4721i	1.58746i	0.608272	+	0.793729i	$0.291863\pi$
$857$	46.4721i	1.58746i	−0.608272	+	0.793729i	$0.708137\pi$
$858$	0	0
$859$	17.0130	0.580477	0.290238	−	0.956954i	$-0.406265\pi$
$859$	17.0130	0.580477	0.290238	+	0.956954i	$0.406265\pi$
$860$	0	0
$861$	−0.450850	−0.0153649
$862$	0	0
$863$	23.5114i	0.800338i	0.916441	+	0.400169i	$0.131049\pi$
$863$	23.5114i	0.800338i	−0.916441	+	0.400169i	$0.868951\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	19.9192i	0.676491i
$868$	0	0
$869$	−18.4164	−0.624734
$870$	0	0
$871$	−4.18774	−0.141896
$872$	0	0
$873$	− 6.23607i	− 0.211059i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	27.9230i	0.942892i	0.881895	+	0.471446i	$0.156268\pi$
$877$	27.9230i	0.942892i	−0.881895	+	0.471446i	$0.843732\pi$
$878$	0	0
$879$	12.5882	0.424591
$880$	0	0
$881$	−45.2492	−1.52449	−0.762243	−	0.647292i	$-0.775902\pi$
$881$	−45.2492	−1.52449	−0.762243	+	0.647292i	$0.775902\pi$
$882$	0	0
$883$	53.9857i	1.81676i	0.418142	+	0.908382i	$0.362682\pi$
$883$	53.9857i	1.81676i	−0.418142	+	0.908382i	$0.637318\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	45.9937i	1.54432i	0.635429	+	0.772159i	$0.280823\pi$
$887$	45.9937i	1.54432i	−0.635429	+	0.772159i	$0.719177\pi$
$888$	0	0
$889$	−5.52786	−0.185399
$890$	0	0
$891$	−4.70228	−0.157532
$892$	0	0
$893$	2.23607i	0.0748272i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 3.41641i	− 0.114071i
$898$	0	0
$899$	−33.0220	−1.10134
$900$	0	0
$901$	1.49342	0.0497531
$902$	0	0
$903$	− 5.87785i	− 0.195603i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 50.9735i	− 1.69255i	−0.532748	−	0.846274i	$-0.678840\pi$
$907$	− 50.9735i	− 1.69255i	0.532748	−	0.846274i	$-0.321160\pi$
$908$	0	0
$909$	−11.3262	−0.375668
$910$	0	0
$911$	−38.0423	−1.26040	−0.630198	−	0.776434i	$-0.717026\pi$
$911$	−38.0423	−1.26040	−0.630198	+	0.776434i	$0.717026\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 11.2574i	− 0.371751i
$918$	0	0
$919$	−4.70228	−0.155114	−0.0775570	−	0.996988i	$-0.524712\pi$
$919$	−4.70228	−0.155114	−0.0775570	+	0.996988i	$0.524712\pi$
$920$	0	0
$921$	−28.2148	−0.929709
$922$	0	0
$923$	− 1.17557i	− 0.0386944i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.52671i	0.115832i
$928$	0	0
$929$	−37.9443	−1.24491	−0.622456	−	0.782655i	$-0.713865\pi$
$929$	−37.9443	−1.24491	−0.622456	+	0.782655i	$0.713865\pi$
$930$	0	0
$931$	32.2299	1.05629
$932$	0	0
$933$	18.8197i	0.616128i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 22.8541i	− 0.746611i	−0.927708	−	0.373305i	$-0.878224\pi$
$937$	− 22.8541i	− 0.746611i	0.927708	−	0.373305i	$-0.121776\pi$
$938$	0	0
$939$	−26.8416	−0.875942
$940$	0	0
$941$	−41.3394	−1.34763	−0.673813	−	0.738902i	$-0.735345\pi$
$941$	−41.3394	−1.34763	−0.673813	+	0.738902i	$0.735345\pi$
$942$	0	0
$943$	4.01623i	0.130786i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 7.99197i	− 0.259704i	−0.991533	−	0.129852i	$-0.958550\pi$
$947$	− 7.99197i	− 0.259704i	0.991533	−	0.129852i	$-0.0414502\pi$
$948$	0	0
$949$	1.00000	0.0324614
$950$	0	0
$951$	−19.9192	−0.645924
$952$	0	0
$953$	− 43.2705i	− 1.40167i	−0.713324	−	0.700835i	$-0.752811\pi$
$953$	− 43.2705i	− 1.40167i	0.713324	−	0.700835i	$-0.247189\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	18.0902i	0.584772i
$958$	0	0
$959$	−1.94260	−0.0627299
$960$	0	0
$961$	12.6180	0.407033
$962$	0	0
$963$	− 1.90211i	− 0.0612947i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	29.9848i	0.964246i	0.876104	+	0.482123i	$0.160134\pi$
$967$	29.9848i	0.964246i	−0.876104	+	0.482123i	$0.839866\pi$
$968$	0	0
$969$	−1.38197	−0.0443951
$970$	0	0
$971$	−8.89002	−0.285294	−0.142647	−	0.989774i	$-0.545561\pi$
$971$	−8.89002	−0.285294	−0.142647	+	0.989774i	$0.545561\pi$
$972$	0	0
$973$	− 5.40325i	− 0.173220i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 35.5623i	− 1.13774i	−0.822428	−	0.568869i	$-0.807381\pi$
$977$	− 35.5623i	− 1.13774i	0.822428	−	0.568869i	$-0.192619\pi$
$978$	0	0
$979$	44.5407	1.42353
$980$	0	0
$981$	19.5623	0.624576
$982$	0	0
$983$	31.0543i	0.990480i	0.868756	+	0.495240i	$0.164920\pi$
$983$	31.0543i	0.990480i	−0.868756	+	0.495240i	$0.835080\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.383516i	0.0122074i
$988$	0	0
$989$	−52.3607	−1.66497
$990$	0	0
$991$	54.1167	1.71907	0.859537	−	0.511073i	$-0.170752\pi$
$991$	54.1167	1.71907	0.859537	+	0.511073i	$0.170752\pi$
$992$	0	0
$993$	− 21.1803i	− 0.672138i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 17.3262i	− 0.548727i	−0.961626	−	0.274364i	$-0.911533\pi$
$997$	− 17.3262i	− 0.548727i	0.961626	−	0.274364i	$-0.0884672\pi$
$998$	0	0
$999$	13.4208	0.424615

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4000.2.c.c.1249.6		8
4.3	odd	2	inner	4000.2.c.c.1249.3		8
5.2	odd	4		4000.2.a.d.1.3	yes	4
5.3	odd	4		4000.2.a.g.1.2	yes	4
5.4	even	2	inner	4000.2.c.c.1249.4		8
20.3	even	4		4000.2.a.g.1.3	yes	4
20.7	even	4		4000.2.a.d.1.2	✓	4
20.19	odd	2	inner	4000.2.c.c.1249.5		8
40.3	even	4		8000.2.a.bf.1.2		4
40.13	odd	4		8000.2.a.bf.1.3		4
40.27	even	4		8000.2.a.bi.1.3		4
40.37	odd	4		8000.2.a.bi.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4000.2.a.d.1.2	✓	4	20.7	even	4
4000.2.a.d.1.3	yes	4	5.2	odd	4
4000.2.a.g.1.2	yes	4	5.3	odd	4
4000.2.a.g.1.3	yes	4	20.3	even	4
4000.2.c.c.1249.3		8	4.3	odd	2	inner
4000.2.c.c.1249.4		8	5.4	even	2	inner
4000.2.c.c.1249.5		8	20.19	odd	2	inner
4000.2.c.c.1249.6		8	1.1	even	1	trivial
8000.2.a.bf.1.2		4	40.3	even	4
8000.2.a.bf.1.3		4	40.13	odd	4
8000.2.a.bi.1.2		4	40.37	odd	4
8000.2.a.bi.1.3		4	40.27	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:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4000.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.17557i q^{3} -0.726543i q^{7} +1.61803 q^{9} +O(q^{10})\)	\(q+1.17557i q^{3} -0.726543i q^{7} +1.61803 q^{9} +3.07768 q^{11} -0.381966i q^{13} +0.236068i q^{17} +4.97980 q^{19} +0.854102 q^{21} -7.60845i q^{23} +5.42882i q^{27} +5.00000 q^{29} -6.60440 q^{31} +3.61803i q^{33} -2.47214i q^{37} +0.449028 q^{39} -0.527864 q^{41} -6.88191i q^{43} +0.449028i q^{47} +6.47214 q^{49} -0.277515 q^{51} -6.32624i q^{53} +5.85410i q^{57} -11.3067 q^{59} +0.854102 q^{61} -1.17557i q^{63} -10.9637i q^{67} +8.94427 q^{69} +3.07768 q^{71} +2.61803i q^{73} -2.23607i q^{77} -5.98385 q^{79} -1.52786 q^{81} +5.87785i q^{87} +14.4721 q^{89} -0.277515 q^{91} -7.76393i q^{93} -3.85410i q^{97} +4.97980 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{9}+O(q^{10}) \) 8 * q + 4 * q^9	\( 8 q + 4 q^{9} - 20 q^{21} + 40 q^{29} - 40 q^{41} + 16 q^{49} - 20 q^{61} - 48 q^{81} + 80 q^{89}+O(q^{100}) \) 8 * q + 4 * q^9 - 20 * q^21 + 40 * q^29 - 40 * q^41 + 16 * q^49 - 20 * q^61 - 48 * q^81 + 80 * q^89

Introduction

L-functions

Modular forms

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Database

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