LMFDB - Embedded newform 5850.2.e.c.5149.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5850,2,Mod(5149,5850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5850, 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("5850.5149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5850.e (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:	$46.7124851824$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1950)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5149.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	5850.5149
Dual form			5850.2.e.c.5149.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +4.00000i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +4.00000i q^{7} -1.00000i q^{8} -4.00000 q^{11} +1.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} -7.00000 q^{19} -4.00000i q^{22} -4.00000i q^{23} -1.00000 q^{26} -4.00000i q^{28} +5.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} +4.00000 q^{34} -9.00000i q^{37} -7.00000i q^{38} +5.00000 q^{41} -10.0000i q^{43} +4.00000 q^{44} +4.00000 q^{46} +3.00000i q^{47} -9.00000 q^{49} -1.00000i q^{52} -9.00000i q^{53} +4.00000 q^{56} +5.00000i q^{58} -6.00000 q^{59} +4.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +7.00000i q^{67} +4.00000i q^{68} +15.0000 q^{71} +12.0000i q^{73} +9.00000 q^{74} +7.00000 q^{76} -16.0000i q^{77} -7.00000 q^{79} +5.00000i q^{82} -6.00000i q^{83} +10.0000 q^{86} +4.00000i q^{88} +14.0000 q^{89} -4.00000 q^{91} +4.00000i q^{92} -3.00000 q^{94} +16.0000i q^{97} -9.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} - 8 q^{11} - 8 q^{14} + 2 q^{16} - 14 q^{19} - 2 q^{26} + 10 q^{29} + 8 q^{31} + 8 q^{34} + 10 q^{41} + 8 q^{44} + 8 q^{46} - 18 q^{49} + 8 q^{56} - 12 q^{59} + 8 q^{61} - 2 q^{64} + 30 q^{71} + 18 q^{74} + 14 q^{76} - 14 q^{79} + 20 q^{86} + 28 q^{89} - 8 q^{91} - 6 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 - 8 * q^11 - 8 * q^14 + 2 * q^16 - 14 * q^19 - 2 * q^26 + 10 * q^29 + 8 * q^31 + 8 * q^34 + 10 * q^41 + 8 * q^44 + 8 * q^46 - 18 * q^49 + 8 * q^56 - 12 * q^59 + 8 * q^61 - 2 * q^64 + 30 * q^71 + 18 * q^74 + 14 * q^76 - 14 * q^79 + 20 * q^86 + 28 * q^89 - 8 * q^91 - 6 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times$.

$n$	$2251$	$3251$	$3277$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	− 4.00000i	− 0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$	− 4.00000i	− 0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$	0	0
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	0	0
$21$	0	0
$22$	− 4.00000i	− 0.852803i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	0	0
$26$	−1.00000	−0.196116
$27$	0	0
$28$	− 4.00000i	− 0.755929i
$29$	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
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	4.00000	0.685994
$35$	0	0
$36$	0	0
$37$	− 9.00000i	− 1.47959i	−0.672832	−	0.739795i	$-0.734922\pi$
$37$	− 9.00000i	− 1.47959i	0.672832	−	0.739795i	$-0.265078\pi$
$38$	− 7.00000i	− 1.13555i
$39$	0	0
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	− 10.0000i	− 1.52499i	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	− 10.0000i	− 1.52499i	0.646997	−	0.762493i	$-0.276025\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	4.00000	0.589768
$47$	3.00000i	0.437595i	0.975770	+	0.218797i	$0.0702134\pi$
$47$	3.00000i	0.437595i	−0.975770	+	0.218797i	$0.929787\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	0	0
$52$	− 1.00000i	− 0.138675i
$53$	− 9.00000i	− 1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$	− 9.00000i	− 1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	0	0
$55$	0	0
$56$	4.00000	0.534522
$57$	0	0
$58$	5.00000i	0.656532i
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	4.00000i	0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	7.00000i	0.855186i	0.903971	+	0.427593i	$0.140638\pi$
$67$	7.00000i	0.855186i	−0.903971	+	0.427593i	$0.859362\pi$
$68$	4.00000i	0.485071i
$69$	0	0
$70$	0	0
$71$	15.0000	1.78017	0.890086	−	0.455792i	$-0.150644\pi$
$71$	15.0000	1.78017	0.890086	+	0.455792i	$0.150644\pi$
$72$	0	0
$73$	12.0000i	1.40449i	0.711934	+	0.702247i	$0.247820\pi$
$73$	12.0000i	1.40449i	−0.711934	+	0.702247i	$0.752180\pi$
$74$	9.00000	1.04623
$75$	0	0
$76$	7.00000	0.802955
$77$	− 16.0000i	− 1.82337i
$78$	0	0
$79$	−7.00000	−0.787562	−0.393781	−	0.919204i	$-0.628833\pi$
$79$	−7.00000	−0.787562	−0.393781	+	0.919204i	$0.628833\pi$
$80$	0	0
$81$	0	0
$82$	5.00000i	0.552158i
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	0	0
$86$	10.0000	1.07833
$87$	0	0
$88$	4.00000i	0.426401i
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	4.00000i	0.417029i
$93$	0	0
$94$	−3.00000	−0.309426
$95$	0	0
$96$	0	0
$97$	16.0000i	1.62455i	0.583272	+	0.812277i	$0.301772\pi$
$97$	16.0000i	1.62455i	−0.583272	+	0.812277i	$0.698228\pi$
$98$	− 9.00000i	− 0.909137i
$99$	0	0
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	9.00000	0.874157
$107$	5.00000i	0.483368i	0.970355	+	0.241684i	$0.0776998\pi$
$107$	5.00000i	0.483368i	−0.970355	+	0.241684i	$0.922300\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	0	0
$112$	4.00000i	0.377964i
$113$	14.0000i	1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$	14.0000i	1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	0	0
$116$	−5.00000	−0.464238
$117$	0	0
$118$	− 6.00000i	− 0.552345i
$119$	16.0000	1.46672
$120$	0	0
$121$	5.00000	0.454545
$122$	4.00000i	0.362143i
$123$	0	0
$124$	−4.00000	−0.359211
$125$	0	0
$126$	0	0
$127$	5.00000i	0.443678i	0.975083	+	0.221839i	$0.0712060\pi$
$127$	5.00000i	0.443678i	−0.975083	+	0.221839i	$0.928794\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−17.0000	−1.48530	−0.742648	−	0.669681i	$-0.766431\pi$
$131$	−17.0000	−1.48530	−0.742648	+	0.669681i	$0.766431\pi$
$132$	0	0
$133$	− 28.0000i	− 2.42791i
$134$	−7.00000	−0.604708
$135$	0	0
$136$	−4.00000	−0.342997
$137$	− 19.0000i	− 1.62328i	−0.584158	−	0.811640i	$-0.698575\pi$
$137$	− 19.0000i	− 1.62328i	0.584158	−	0.811640i	$-0.301425\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	15.0000i	1.25877i
$143$	− 4.00000i	− 0.334497i
$144$	0	0
$145$	0	0
$146$	−12.0000	−0.993127
$147$	0	0
$148$	9.00000i	0.739795i
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	7.00000i	0.567775i
$153$	0	0
$154$	16.0000	1.28932
$155$	0	0
$156$	0	0
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	− 7.00000i	− 0.556890i
$159$	0	0
$160$	0	0
$161$	16.0000	1.26098
$162$	0	0
$163$	− 24.0000i	− 1.87983i	−0.341415	−	0.939913i	$-0.610906\pi$
$163$	− 24.0000i	− 1.87983i	0.341415	−	0.939913i	$-0.389094\pi$
$164$	−5.00000	−0.390434
$165$	0	0
$166$	6.00000	0.465690
$167$	9.00000i	0.696441i	0.937413	+	0.348220i	$0.113214\pi$
$167$	9.00000i	0.696441i	−0.937413	+	0.348220i	$0.886786\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	10.0000i	0.762493i
$173$	− 13.0000i	− 0.988372i	−0.869356	−	0.494186i	$-0.835466\pi$
$173$	− 13.0000i	− 0.988372i	0.869356	−	0.494186i	$-0.164534\pi$
$174$	0	0
$175$	0	0
$176$	−4.00000	−0.301511
$177$	0	0
$178$	14.0000i	1.04934i
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−4.00000	−0.297318	−0.148659	−	0.988889i	$-0.547496\pi$
$181$	−4.00000	−0.297318	−0.148659	+	0.988889i	$0.547496\pi$
$182$	− 4.00000i	− 0.296500i
$183$	0	0
$184$	−4.00000	−0.294884
$185$	0	0
$186$	0	0
$187$	16.0000i	1.17004i
$188$	− 3.00000i	− 0.218797i
$189$	0	0
$190$	0	0
$191$	−10.0000	−0.723575	−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000	−0.723575	−0.361787	+	0.932261i	$0.617833\pi$
$192$	0	0
$193$	− 12.0000i	− 0.863779i	−0.901927	−	0.431889i	$-0.857847\pi$
$193$	− 12.0000i	− 0.863779i	0.901927	−	0.431889i	$-0.142153\pi$
$194$	−16.0000	−1.14873
$195$	0	0
$196$	9.00000	0.642857
$197$	− 16.0000i	− 1.13995i	−0.821661	−	0.569976i	$-0.806952\pi$
$197$	− 16.0000i	− 1.13995i	0.821661	−	0.569976i	$-0.193048\pi$
$198$	0	0
$199$	3.00000	0.212664	0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000	0.212664	0.106332	+	0.994331i	$0.466089\pi$
$200$	0	0
$201$	0	0
$202$	10.0000i	0.703598i
$203$	20.0000i	1.40372i
$204$	0	0
$205$	0	0
$206$	−8.00000	−0.557386
$207$	0	0
$208$	1.00000i	0.0693375i
$209$	28.0000	1.93680
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	9.00000i	0.618123i
$213$	0	0
$214$	−5.00000	−0.341793
$215$	0	0
$216$	0	0
$217$	16.0000i	1.08615i
$218$	11.0000i	0.745014i
$219$	0	0
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	− 2.00000i	− 0.133930i	−0.997755	−	0.0669650i	$-0.978668\pi$
$223$	− 2.00000i	− 0.133930i	0.997755	−	0.0669650i	$-0.0213316\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	−14.0000	−0.931266
$227$	6.00000i	0.398234i	0.979976	+	0.199117i	$0.0638074\pi$
$227$	6.00000i	0.398234i	−0.979976	+	0.199117i	$0.936193\pi$
$228$	0	0
$229$	17.0000	1.12339	0.561696	−	0.827344i	$-0.310149\pi$
$229$	17.0000	1.12339	0.561696	+	0.827344i	$0.310149\pi$
$230$	0	0
$231$	0	0
$232$	− 5.00000i	− 0.328266i
$233$	− 8.00000i	− 0.524097i	−0.965055	−	0.262049i	$-0.915602\pi$
$233$	− 8.00000i	− 0.524097i	0.965055	−	0.262049i	$-0.0843981\pi$
$234$	0	0
$235$	0	0
$236$	6.00000	0.390567
$237$	0	0
$238$	16.0000i	1.03713i
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	5.00000i	0.321412i
$243$	0	0
$244$	−4.00000	−0.256074
$245$	0	0
$246$	0	0
$247$	− 7.00000i	− 0.445399i
$248$	− 4.00000i	− 0.254000i
$249$	0	0
$250$	0	0
$251$	5.00000	0.315597	0.157799	−	0.987471i	$-0.449560\pi$
$251$	5.00000	0.315597	0.157799	+	0.987471i	$0.449560\pi$
$252$	0	0
$253$	16.0000i	1.00591i
$254$	−5.00000	−0.313728
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.00000i	0.499026i	0.968371	+	0.249513i	$0.0802706\pi$
$257$	8.00000i	0.499026i	−0.968371	+	0.249513i	$0.919729\pi$
$258$	0	0
$259$	36.0000	2.23693
$260$	0	0
$261$	0	0
$262$	− 17.0000i	− 1.05026i
$263$	− 30.0000i	− 1.84988i	−0.380114	−	0.924940i	$-0.624115\pi$
$263$	− 30.0000i	− 1.84988i	0.380114	−	0.924940i	$-0.375885\pi$
$264$	0	0
$265$	0	0
$266$	28.0000	1.71679
$267$	0	0
$268$	− 7.00000i	− 0.427593i
$269$	5.00000	0.304855	0.152428	−	0.988315i	$-0.451291\pi$
$269$	5.00000	0.304855	0.152428	+	0.988315i	$0.451291\pi$
$270$	0	0
$271$	32.0000	1.94386	0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000	1.94386	0.971931	+	0.235267i	$0.0755965\pi$
$272$	− 4.00000i	− 0.242536i
$273$	0	0
$274$	19.0000	1.14783
$275$	0	0
$276$	0	0
$277$	− 26.0000i	− 1.56219i	−0.624413	−	0.781094i	$-0.714662\pi$
$277$	− 26.0000i	− 1.56219i	0.624413	−	0.781094i	$-0.285338\pi$
$278$	4.00000i	0.239904i
$279$	0	0
$280$	0	0
$281$	−17.0000	−1.01413	−0.507067	−	0.861906i	$-0.669271\pi$
$281$	−17.0000	−1.01413	−0.507067	+	0.861906i	$0.669271\pi$
$282$	0	0
$283$	8.00000i	0.475551i	0.971320	+	0.237775i	$0.0764182\pi$
$283$	8.00000i	0.475551i	−0.971320	+	0.237775i	$0.923582\pi$
$284$	−15.0000	−0.890086
$285$	0	0
$286$	4.00000	0.236525
$287$	20.0000i	1.18056i
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	− 12.0000i	− 0.702247i
$293$	12.0000i	0.701047i	0.936554	+	0.350524i	$0.113996\pi$
$293$	12.0000i	0.701047i	−0.936554	+	0.350524i	$0.886004\pi$
$294$	0	0
$295$	0	0
$296$	−9.00000	−0.523114
$297$	0	0
$298$	− 4.00000i	− 0.231714i
$299$	4.00000	0.231326
$300$	0	0
$301$	40.0000	2.30556
$302$	− 4.00000i	− 0.230174i
$303$	0	0
$304$	−7.00000	−0.401478
$305$	0	0
$306$	0	0
$307$	− 21.0000i	− 1.19853i	−0.800549	−	0.599267i	$-0.795459\pi$
$307$	− 21.0000i	− 1.19853i	0.800549	−	0.599267i	$-0.204541\pi$
$308$	16.0000i	0.911685i
$309$	0	0
$310$	0	0
$311$	−2.00000	−0.113410	−0.0567048	−	0.998391i	$-0.518059\pi$
$311$	−2.00000	−0.113410	−0.0567048	+	0.998391i	$0.518059\pi$
$312$	0	0
$313$	− 23.0000i	− 1.30004i	−0.759918	−	0.650018i	$-0.774761\pi$
$313$	− 23.0000i	− 1.30004i	0.759918	−	0.650018i	$-0.225239\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	7.00000	0.393781
$317$	− 28.0000i	− 1.57264i	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	− 28.0000i	− 1.57264i	0.617822	−	0.786318i	$-0.288015\pi$
$318$	0	0
$319$	−20.0000	−1.11979
$320$	0	0
$321$	0	0
$322$	16.0000i	0.891645i
$323$	28.0000i	1.55796i
$324$	0	0
$325$	0	0
$326$	24.0000	1.32924
$327$	0	0
$328$	− 5.00000i	− 0.276079i
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	6.00000i	0.329293i
$333$	0	0
$334$	−9.00000	−0.492458
$335$	0	0
$336$	0	0
$337$	34.0000i	1.85210i	0.377403	+	0.926049i	$0.376817\pi$
$337$	34.0000i	1.85210i	−0.377403	+	0.926049i	$0.623183\pi$
$338$	− 1.00000i	− 0.0543928i
$339$	0	0
$340$	0	0
$341$	−16.0000	−0.866449
$342$	0	0
$343$	− 8.00000i	− 0.431959i
$344$	−10.0000	−0.539164
$345$	0	0
$346$	13.0000	0.698884
$347$	11.0000i	0.590511i	0.955418	+	0.295255i	$0.0954048\pi$
$347$	11.0000i	0.590511i	−0.955418	+	0.295255i	$0.904595\pi$
$348$	0	0
$349$	34.0000	1.81998	0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000	1.81998	0.909989	+	0.414632i	$0.136090\pi$
$350$	0	0
$351$	0	0
$352$	− 4.00000i	− 0.213201i
$353$	21.0000i	1.11772i	0.829263	+	0.558859i	$0.188761\pi$
$353$	21.0000i	1.11772i	−0.829263	+	0.558859i	$0.811239\pi$
$354$	0	0
$355$	0	0
$356$	−14.0000	−0.741999
$357$	0	0
$358$	− 4.00000i	− 0.211407i
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	− 4.00000i	− 0.210235i
$363$	0	0
$364$	4.00000	0.209657
$365$	0	0
$366$	0	0
$367$	− 7.00000i	− 0.365397i	−0.983169	−	0.182699i	$-0.941517\pi$
$367$	− 7.00000i	− 0.365397i	0.983169	−	0.182699i	$-0.0584832\pi$
$368$	− 4.00000i	− 0.208514i
$369$	0	0
$370$	0	0
$371$	36.0000	1.86903
$372$	0	0
$373$	22.0000i	1.13912i	0.821951	+	0.569558i	$0.192886\pi$
$373$	22.0000i	1.13912i	−0.821951	+	0.569558i	$0.807114\pi$
$374$	−16.0000	−0.827340
$375$	0	0
$376$	3.00000	0.154713
$377$	5.00000i	0.257513i
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	0	0
$382$	− 10.0000i	− 0.511645i
$383$	− 5.00000i	− 0.255488i	−0.991807	−	0.127744i	$-0.959226\pi$
$383$	− 5.00000i	− 0.255488i	0.991807	−	0.127744i	$-0.0407736\pi$
$384$	0	0
$385$	0	0
$386$	12.0000	0.610784
$387$	0	0
$388$	− 16.0000i	− 0.812277i
$389$	5.00000	0.253510	0.126755	−	0.991934i	$-0.459544\pi$
$389$	5.00000	0.253510	0.126755	+	0.991934i	$0.459544\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	9.00000i	0.454569i
$393$	0	0
$394$	16.0000	0.806068
$395$	0	0
$396$	0	0
$397$	15.0000i	0.752828i	0.926451	+	0.376414i	$0.122843\pi$
$397$	15.0000i	0.752828i	−0.926451	+	0.376414i	$0.877157\pi$
$398$	3.00000i	0.150376i
$399$	0	0
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	4.00000i	0.199254i
$404$	−10.0000	−0.497519
$405$	0	0
$406$	−20.0000	−0.992583
$407$	36.0000i	1.78445i
$408$	0	0
$409$	24.0000	1.18672	0.593362	−	0.804936i	$-0.297800\pi$
$409$	24.0000	1.18672	0.593362	+	0.804936i	$0.297800\pi$
$410$	0	0
$411$	0	0
$412$	− 8.00000i	− 0.394132i
$413$	− 24.0000i	− 1.18096i
$414$	0	0
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	28.0000i	1.36952i
$419$	23.0000	1.12362	0.561812	−	0.827265i	$-0.310105\pi$
$419$	23.0000	1.12362	0.561812	+	0.827265i	$0.310105\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	− 16.0000i	− 0.778868i
$423$	0	0
$424$	−9.00000	−0.437079
$425$	0	0
$426$	0	0
$427$	16.0000i	0.774294i
$428$	− 5.00000i	− 0.241684i
$429$	0	0
$430$	0	0
$431$	−19.0000	−0.915198	−0.457599	−	0.889159i	$-0.651290\pi$
$431$	−19.0000	−0.915198	−0.457599	+	0.889159i	$0.651290\pi$
$432$	0	0
$433$	− 19.0000i	− 0.913082i	−0.889702	−	0.456541i	$-0.849088\pi$
$433$	− 19.0000i	− 0.913082i	0.889702	−	0.456541i	$-0.150912\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	−11.0000	−0.526804
$437$	28.0000i	1.33942i
$438$	0	0
$439$	7.00000	0.334092	0.167046	−	0.985949i	$-0.446577\pi$
$439$	7.00000	0.334092	0.167046	+	0.985949i	$0.446577\pi$
$440$	0	0
$441$	0	0
$442$	4.00000i	0.190261i
$443$	− 25.0000i	− 1.18779i	−0.804544	−	0.593893i	$-0.797590\pi$
$443$	− 25.0000i	− 1.18779i	0.804544	−	0.593893i	$-0.202410\pi$
$444$	0	0
$445$	0	0
$446$	2.00000	0.0947027
$447$	0	0
$448$	− 4.00000i	− 0.188982i
$449$	−27.0000	−1.27421	−0.637104	−	0.770778i	$-0.719868\pi$
$449$	−27.0000	−1.27421	−0.637104	+	0.770778i	$0.719868\pi$
$450$	0	0
$451$	−20.0000	−0.941763
$452$	− 14.0000i	− 0.658505i
$453$	0	0
$454$	−6.00000	−0.281594
$455$	0	0
$456$	0	0
$457$	− 4.00000i	− 0.187112i	−0.995614	−	0.0935561i	$-0.970177\pi$
$457$	− 4.00000i	− 0.187112i	0.995614	−	0.0935561i	$-0.0298234\pi$
$458$	17.0000i	0.794358i
$459$	0	0
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	0	0
$463$	− 26.0000i	− 1.20832i	−0.796862	−	0.604161i	$-0.793508\pi$
$463$	− 26.0000i	− 1.20832i	0.796862	−	0.604161i	$-0.206492\pi$
$464$	5.00000	0.232119
$465$	0	0
$466$	8.00000	0.370593
$467$	3.00000i	0.138823i	0.997588	+	0.0694117i	$0.0221122\pi$
$467$	3.00000i	0.138823i	−0.997588	+	0.0694117i	$0.977888\pi$
$468$	0	0
$469$	−28.0000	−1.29292
$470$	0	0
$471$	0	0
$472$	6.00000i	0.276172i
$473$	40.0000i	1.83920i
$474$	0	0
$475$	0	0
$476$	−16.0000	−0.733359
$477$	0	0
$478$	16.0000i	0.731823i
$479$	15.0000	0.685367	0.342684	−	0.939451i	$-0.388664\pi$
$479$	15.0000	0.685367	0.342684	+	0.939451i	$0.388664\pi$
$480$	0	0
$481$	9.00000	0.410365
$482$	8.00000i	0.364390i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	0	0
$487$	− 22.0000i	− 0.996915i	−0.866914	−	0.498458i	$-0.833900\pi$
$487$	− 22.0000i	− 0.996915i	0.866914	−	0.498458i	$-0.166100\pi$
$488$	− 4.00000i	− 0.181071i
$489$	0	0
$490$	0	0
$491$	40.0000	1.80517	0.902587	−	0.430507i	$-0.141665\pi$
$491$	40.0000	1.80517	0.902587	+	0.430507i	$0.141665\pi$
$492$	0	0
$493$	− 20.0000i	− 0.900755i
$494$	7.00000	0.314945
$495$	0	0
$496$	4.00000	0.179605
$497$	60.0000i	2.69137i
$498$	0	0
$499$	23.0000	1.02962	0.514811	−	0.857304i	$-0.327862\pi$
$499$	23.0000	1.02962	0.514811	+	0.857304i	$0.327862\pi$
$500$	0	0
$501$	0	0
$502$	5.00000i	0.223161i
$503$	− 36.0000i	− 1.60516i	−0.596544	−	0.802580i	$-0.703460\pi$
$503$	− 36.0000i	− 1.60516i	0.596544	−	0.802580i	$-0.296540\pi$
$504$	0	0
$505$	0	0
$506$	−16.0000	−0.711287
$507$	0	0
$508$	− 5.00000i	− 0.221839i
$509$	−4.00000	−0.177297	−0.0886484	−	0.996063i	$-0.528255\pi$
$509$	−4.00000	−0.177297	−0.0886484	+	0.996063i	$0.528255\pi$
$510$	0	0
$511$	−48.0000	−2.12339
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	−8.00000	−0.352865
$515$	0	0
$516$	0	0
$517$	− 12.0000i	− 0.527759i
$518$	36.0000i	1.58175i
$519$	0	0
$520$	0	0
$521$	−36.0000	−1.57719	−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000	−1.57719	−0.788594	+	0.614914i	$0.789191\pi$
$522$	0	0
$523$	− 14.0000i	− 0.612177i	−0.952003	−	0.306089i	$-0.900980\pi$
$523$	− 14.0000i	− 0.612177i	0.952003	−	0.306089i	$-0.0990204\pi$
$524$	17.0000	0.742648
$525$	0	0
$526$	30.0000	1.30806
$527$	− 16.0000i	− 0.696971i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	28.0000i	1.21395i
$533$	5.00000i	0.216574i
$534$	0	0
$535$	0	0
$536$	7.00000	0.302354
$537$	0	0
$538$	5.00000i	0.215565i
$539$	36.0000	1.55063
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	32.0000i	1.37452i
$543$	0	0
$544$	4.00000	0.171499
$545$	0	0
$546$	0	0
$547$	− 6.00000i	− 0.256541i	−0.991739	−	0.128271i	$-0.959057\pi$
$547$	− 6.00000i	− 0.256541i	0.991739	−	0.128271i	$-0.0409426\pi$
$548$	19.0000i	0.811640i
$549$	0	0
$550$	0	0
$551$	−35.0000	−1.49105
$552$	0	0
$553$	− 28.0000i	− 1.19068i
$554$	26.0000	1.10463
$555$	0	0
$556$	−4.00000	−0.169638
$557$	− 38.0000i	− 1.61011i	−0.593199	−	0.805056i	$-0.702135\pi$
$557$	− 38.0000i	− 1.61011i	0.593199	−	0.805056i	$-0.297865\pi$
$558$	0	0
$559$	10.0000	0.422955
$560$	0	0
$561$	0	0
$562$	− 17.0000i	− 0.717102i
$563$	− 39.0000i	− 1.64365i	−0.569737	−	0.821827i	$-0.692955\pi$
$563$	− 39.0000i	− 1.64365i	0.569737	−	0.821827i	$-0.307045\pi$
$564$	0	0
$565$	0	0
$566$	−8.00000	−0.336265
$567$	0	0
$568$	− 15.0000i	− 0.629386i
$569$	8.00000	0.335377	0.167689	−	0.985840i	$-0.446370\pi$
$569$	8.00000	0.335377	0.167689	+	0.985840i	$0.446370\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	4.00000i	0.167248i
$573$	0	0
$574$	−20.0000	−0.834784
$575$	0	0
$576$	0	0
$577$	8.00000i	0.333044i	0.986038	+	0.166522i	$0.0532537\pi$
$577$	8.00000i	0.333044i	−0.986038	+	0.166522i	$0.946746\pi$
$578$	1.00000i	0.0415945i
$579$	0	0
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	36.0000i	1.49097i
$584$	12.0000	0.496564
$585$	0	0
$586$	−12.0000	−0.495715
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	0	0
$589$	−28.0000	−1.15372
$590$	0	0
$591$	0	0
$592$	− 9.00000i	− 0.369898i
$593$	− 29.0000i	− 1.19089i	−0.803397	−	0.595444i	$-0.796976\pi$
$593$	− 29.0000i	− 1.19089i	0.803397	−	0.595444i	$-0.203024\pi$
$594$	0	0
$595$	0	0
$596$	4.00000	0.163846
$597$	0	0
$598$	4.00000i	0.163572i
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	−3.00000	−0.122373	−0.0611863	−	0.998126i	$-0.519488\pi$
$601$	−3.00000	−0.122373	−0.0611863	+	0.998126i	$0.519488\pi$
$602$	40.0000i	1.63028i
$603$	0	0
$604$	4.00000	0.162758
$605$	0	0
$606$	0	0
$607$	3.00000i	0.121766i	0.998145	+	0.0608831i	$0.0193917\pi$
$607$	3.00000i	0.121766i	−0.998145	+	0.0608831i	$0.980608\pi$
$608$	− 7.00000i	− 0.283887i
$609$	0	0
$610$	0	0
$611$	−3.00000	−0.121367
$612$	0	0
$613$	2.00000i	0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$	2.00000i	0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$	21.0000	0.847491
$615$	0	0
$616$	−16.0000	−0.644658
$617$	7.00000i	0.281809i	0.990023	+	0.140905i	$0.0450011\pi$
$617$	7.00000i	0.281809i	−0.990023	+	0.140905i	$0.954999\pi$
$618$	0	0
$619$	−8.00000	−0.321547	−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000	−0.321547	−0.160774	+	0.986991i	$0.551399\pi$
$620$	0	0
$621$	0	0
$622$	− 2.00000i	− 0.0801927i
$623$	56.0000i	2.24359i
$624$	0	0
$625$	0	0
$626$	23.0000	0.919265
$627$	0	0
$628$	− 14.0000i	− 0.558661i
$629$	−36.0000	−1.43541
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	7.00000i	0.278445i
$633$	0	0
$634$	28.0000	1.11202
$635$	0	0
$636$	0	0
$637$	− 9.00000i	− 0.356593i
$638$	− 20.0000i	− 0.791808i
$639$	0	0
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	− 5.00000i	− 0.197181i	−0.995128	−	0.0985904i	$-0.968567\pi$
$643$	− 5.00000i	− 0.197181i	0.995128	−	0.0985904i	$-0.0314334\pi$
$644$	−16.0000	−0.630488
$645$	0	0
$646$	−28.0000	−1.10165
$647$	− 36.0000i	− 1.41531i	−0.706560	−	0.707653i	$-0.749754\pi$
$647$	− 36.0000i	− 1.41531i	0.706560	−	0.707653i	$-0.250246\pi$
$648$	0	0
$649$	24.0000	0.942082
$650$	0	0
$651$	0	0
$652$	24.0000i	0.939913i
$653$	10.0000i	0.391330i	0.980671	+	0.195665i	$0.0626866\pi$
$653$	10.0000i	0.391330i	−0.980671	+	0.195665i	$0.937313\pi$
$654$	0	0
$655$	0	0
$656$	5.00000	0.195217
$657$	0	0
$658$	− 12.0000i	− 0.467809i
$659$	33.0000	1.28550	0.642749	−	0.766077i	$-0.277794\pi$
$659$	33.0000	1.28550	0.642749	+	0.766077i	$0.277794\pi$
$660$	0	0
$661$	−7.00000	−0.272268	−0.136134	−	0.990690i	$-0.543468\pi$
$661$	−7.00000	−0.272268	−0.136134	+	0.990690i	$0.543468\pi$
$662$	− 28.0000i	− 1.08825i
$663$	0	0
$664$	−6.00000	−0.232845
$665$	0	0
$666$	0	0
$667$	− 20.0000i	− 0.774403i
$668$	− 9.00000i	− 0.348220i
$669$	0	0
$670$	0	0
$671$	−16.0000	−0.617673
$672$	0	0
$673$	19.0000i	0.732396i	0.930537	+	0.366198i	$0.119341\pi$
$673$	19.0000i	0.732396i	−0.930537	+	0.366198i	$0.880659\pi$
$674$	−34.0000	−1.30963
$675$	0	0
$676$	1.00000	0.0384615
$677$	− 14.0000i	− 0.538064i	−0.963131	−	0.269032i	$-0.913296\pi$
$677$	− 14.0000i	− 0.538064i	0.963131	−	0.269032i	$-0.0867037\pi$
$678$	0	0
$679$	−64.0000	−2.45609
$680$	0	0
$681$	0	0
$682$	− 16.0000i	− 0.612672i
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	8.00000	0.305441
$687$	0	0
$688$	− 10.0000i	− 0.381246i
$689$	9.00000	0.342873
$690$	0	0
$691$	3.00000	0.114125	0.0570627	−	0.998371i	$-0.481827\pi$
$691$	3.00000	0.114125	0.0570627	+	0.998371i	$0.481827\pi$
$692$	13.0000i	0.494186i
$693$	0	0
$694$	−11.0000	−0.417554
$695$	0	0
$696$	0	0
$697$	− 20.0000i	− 0.757554i
$698$	34.0000i	1.28692i
$699$	0	0
$700$	0	0
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	0	0
$703$	63.0000i	2.37609i
$704$	4.00000	0.150756
$705$	0	0
$706$	−21.0000	−0.790345
$707$	40.0000i	1.50435i
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	0	0
$711$	0	0
$712$	− 14.0000i	− 0.524672i
$713$	− 16.0000i	− 0.599205i
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	0	0
$718$	− 15.0000i	− 0.559795i
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	0	0
$721$	−32.0000	−1.19174
$722$	30.0000i	1.11648i
$723$	0	0
$724$	4.00000	0.148659
$725$	0	0
$726$	0	0
$727$	− 28.0000i	− 1.03846i	−0.854634	−	0.519231i	$-0.826218\pi$
$727$	− 28.0000i	− 1.03846i	0.854634	−	0.519231i	$-0.173782\pi$
$728$	4.00000i	0.148250i
$729$	0	0
$730$	0	0
$731$	−40.0000	−1.47945
$732$	0	0
$733$	− 3.00000i	− 0.110808i	−0.998464	−	0.0554038i	$-0.982355\pi$
$733$	− 3.00000i	− 0.110808i	0.998464	−	0.0554038i	$-0.0176446\pi$
$734$	7.00000	0.258375
$735$	0	0
$736$	4.00000	0.147442
$737$	− 28.0000i	− 1.03139i
$738$	0	0
$739$	37.0000	1.36107	0.680534	−	0.732717i	$-0.261748\pi$
$739$	37.0000	1.36107	0.680534	+	0.732717i	$0.261748\pi$
$740$	0	0
$741$	0	0
$742$	36.0000i	1.32160i
$743$	21.0000i	0.770415i	0.922830	+	0.385208i	$0.125870\pi$
$743$	21.0000i	0.770415i	−0.922830	+	0.385208i	$0.874130\pi$
$744$	0	0
$745$	0	0
$746$	−22.0000	−0.805477
$747$	0	0
$748$	− 16.0000i	− 0.585018i
$749$	−20.0000	−0.730784
$750$	0	0
$751$	−23.0000	−0.839282	−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000	−0.839282	−0.419641	+	0.907690i	$0.637844\pi$
$752$	3.00000i	0.109399i
$753$	0	0
$754$	−5.00000	−0.182089
$755$	0	0
$756$	0	0
$757$	− 8.00000i	− 0.290765i	−0.989376	−	0.145382i	$-0.953559\pi$
$757$	− 8.00000i	− 0.290765i	0.989376	−	0.145382i	$-0.0464413\pi$
$758$	20.0000i	0.726433i
$759$	0	0
$760$	0	0
$761$	−35.0000	−1.26875	−0.634375	−	0.773026i	$-0.718742\pi$
$761$	−35.0000	−1.26875	−0.634375	+	0.773026i	$0.718742\pi$
$762$	0	0
$763$	44.0000i	1.59291i
$764$	10.0000	0.361787
$765$	0	0
$766$	5.00000	0.180657
$767$	− 6.00000i	− 0.216647i
$768$	0	0
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	0	0
$772$	12.0000i	0.431889i
$773$	8.00000i	0.287740i	0.989597	+	0.143870i	$0.0459547\pi$
$773$	8.00000i	0.287740i	−0.989597	+	0.143870i	$0.954045\pi$
$774$	0	0
$775$	0	0
$776$	16.0000	0.574367
$777$	0	0
$778$	5.00000i	0.179259i
$779$	−35.0000	−1.25401
$780$	0	0
$781$	−60.0000	−2.14697
$782$	− 16.0000i	− 0.572159i
$783$	0	0
$784$	−9.00000	−0.321429
$785$	0	0
$786$	0	0
$787$	32.0000i	1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$	32.0000i	1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$	16.0000i	0.569976i
$789$	0	0
$790$	0	0
$791$	−56.0000	−1.99113
$792$	0	0
$793$	4.00000i	0.142044i
$794$	−15.0000	−0.532330
$795$	0	0
$796$	−3.00000	−0.106332
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	0	0
$801$	0	0
$802$	30.0000i	1.05934i
$803$	− 48.0000i	− 1.69388i
$804$	0	0
$805$	0	0
$806$	−4.00000	−0.140894
$807$	0	0
$808$	− 10.0000i	− 0.351799i
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	− 20.0000i	− 0.701862i
$813$	0	0
$814$	−36.0000	−1.26180
$815$	0	0
$816$	0	0
$817$	70.0000i	2.44899i
$818$	24.0000i	0.839140i
$819$	0	0
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	0	0
$823$	− 19.0000i	− 0.662298i	−0.943578	−	0.331149i	$-0.892564\pi$
$823$	− 19.0000i	− 0.662298i	0.943578	−	0.331149i	$-0.107436\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	24.0000	0.835067
$827$	− 4.00000i	− 0.139094i	−0.997579	−	0.0695468i	$-0.977845\pi$
$827$	− 4.00000i	− 0.139094i	0.997579	−	0.0695468i	$-0.0221553\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	0	0
$832$	− 1.00000i	− 0.0346688i
$833$	36.0000i	1.24733i
$834$	0	0
$835$	0	0
$836$	−28.0000	−0.968400
$837$	0	0
$838$	23.0000i	0.794522i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	14.0000i	0.482472i
$843$	0	0
$844$	16.0000	0.550743
$845$	0	0
$846$	0	0
$847$	20.0000i	0.687208i
$848$	− 9.00000i	− 0.309061i
$849$	0	0
$850$	0	0
$851$	−36.0000	−1.23406
$852$	0	0
$853$	19.0000i	0.650548i	0.945620	+	0.325274i	$0.105456\pi$
$853$	19.0000i	0.650548i	−0.945620	+	0.325274i	$0.894544\pi$
$854$	−16.0000	−0.547509
$855$	0	0
$856$	5.00000	0.170896
$857$	− 50.0000i	− 1.70797i	−0.520300	−	0.853984i	$-0.674180\pi$
$857$	− 50.0000i	− 1.70797i	0.520300	−	0.853984i	$-0.325820\pi$
$858$	0	0
$859$	2.00000	0.0682391	0.0341196	−	0.999418i	$-0.489137\pi$
$859$	2.00000	0.0682391	0.0341196	+	0.999418i	$0.489137\pi$
$860$	0	0
$861$	0	0
$862$	− 19.0000i	− 0.647143i
$863$	− 39.0000i	− 1.32758i	−0.747921	−	0.663788i	$-0.768948\pi$
$863$	− 39.0000i	− 1.32758i	0.747921	−	0.663788i	$-0.231052\pi$
$864$	0	0
$865$	0	0
$866$	19.0000	0.645646
$867$	0	0
$868$	− 16.0000i	− 0.543075i
$869$	28.0000	0.949835
$870$	0	0
$871$	−7.00000	−0.237186
$872$	− 11.0000i	− 0.372507i
$873$	0	0
$874$	−28.0000	−0.947114
$875$	0	0
$876$	0	0
$877$	− 43.0000i	− 1.45201i	−0.687691	−	0.726003i	$-0.741376\pi$
$877$	− 43.0000i	− 1.45201i	0.687691	−	0.726003i	$-0.258624\pi$
$878$	7.00000i	0.236239i
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	40.0000i	1.34611i	0.739594	+	0.673054i	$0.235018\pi$
$883$	40.0000i	1.34611i	−0.739594	+	0.673054i	$0.764982\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	25.0000	0.839891
$887$	− 36.0000i	− 1.20876i	−0.796696	−	0.604381i	$-0.793421\pi$
$887$	− 36.0000i	− 1.20876i	0.796696	−	0.604381i	$-0.206579\pi$
$888$	0	0
$889$	−20.0000	−0.670778
$890$	0	0
$891$	0	0
$892$	2.00000i	0.0669650i
$893$	− 21.0000i	− 0.702738i
$894$	0	0
$895$	0	0
$896$	4.00000	0.133631
$897$	0	0
$898$	− 27.0000i	− 0.901002i
$899$	20.0000	0.667037
$900$	0	0
$901$	−36.0000	−1.19933
$902$	− 20.0000i	− 0.665927i
$903$	0	0
$904$	14.0000	0.465633
$905$	0	0
$906$	0	0
$907$	− 50.0000i	− 1.66022i	−0.557598	−	0.830111i	$-0.688277\pi$
$907$	− 50.0000i	− 1.66022i	0.557598	−	0.830111i	$-0.311723\pi$
$908$	− 6.00000i	− 0.199117i
$909$	0	0
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	0	0
$913$	24.0000i	0.794284i
$914$	4.00000	0.132308
$915$	0	0
$916$	−17.0000	−0.561696
$917$	− 68.0000i	− 2.24556i
$918$	0	0
$919$	−29.0000	−0.956622	−0.478311	−	0.878191i	$-0.658751\pi$
$919$	−29.0000	−0.956622	−0.478311	+	0.878191i	$0.658751\pi$
$920$	0	0
$921$	0	0
$922$	24.0000i	0.790398i
$923$	15.0000i	0.493731i
$924$	0	0
$925$	0	0
$926$	26.0000	0.854413
$927$	0	0
$928$	5.00000i	0.164133i
$929$	11.0000	0.360898	0.180449	−	0.983584i	$-0.442245\pi$
$929$	11.0000	0.360898	0.180449	+	0.983584i	$0.442245\pi$
$930$	0	0
$931$	63.0000	2.06474
$932$	8.00000i	0.262049i
$933$	0	0
$934$	−3.00000	−0.0981630
$935$	0	0
$936$	0	0
$937$	22.0000i	0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$	22.0000i	0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$	− 28.0000i	− 0.914232i
$939$	0	0
$940$	0	0
$941$	−52.0000	−1.69515	−0.847576	−	0.530674i	$-0.821939\pi$
$941$	−52.0000	−1.69515	−0.847576	+	0.530674i	$0.821939\pi$
$942$	0	0
$943$	− 20.0000i	− 0.651290i
$944$	−6.00000	−0.195283
$945$	0	0
$946$	−40.0000	−1.30051
$947$	− 8.00000i	− 0.259965i	−0.991516	−	0.129983i	$-0.958508\pi$
$947$	− 8.00000i	− 0.259965i	0.991516	−	0.129983i	$-0.0414921\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	0	0
$951$	0	0
$952$	− 16.0000i	− 0.518563i
$953$	− 36.0000i	− 1.16615i	−0.812417	−	0.583077i	$-0.801849\pi$
$953$	− 36.0000i	− 1.16615i	0.812417	−	0.583077i	$-0.198151\pi$
$954$	0	0
$955$	0	0
$956$	−16.0000	−0.517477
$957$	0	0
$958$	15.0000i	0.484628i
$959$	76.0000	2.45417
$960$	0	0
$961$	−15.0000	−0.483871
$962$	9.00000i	0.290172i
$963$	0	0
$964$	−8.00000	−0.257663
$965$	0	0
$966$	0	0
$967$	44.0000i	1.41494i	0.706741	+	0.707472i	$0.250165\pi$
$967$	44.0000i	1.41494i	−0.706741	+	0.707472i	$0.749835\pi$
$968$	− 5.00000i	− 0.160706i
$969$	0	0
$970$	0	0
$971$	21.0000	0.673922	0.336961	−	0.941519i	$-0.390601\pi$
$971$	21.0000	0.673922	0.336961	+	0.941519i	$0.390601\pi$
$972$	0	0
$973$	16.0000i	0.512936i
$974$	22.0000	0.704925
$975$	0	0
$976$	4.00000	0.128037
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	0	0
$979$	−56.0000	−1.78977
$980$	0	0
$981$	0	0
$982$	40.0000i	1.27645i
$983$	48.0000i	1.53096i	0.643458	+	0.765481i	$0.277499\pi$
$983$	48.0000i	1.53096i	−0.643458	+	0.765481i	$0.722501\pi$
$984$	0	0
$985$	0	0
$986$	20.0000	0.636930
$987$	0	0
$988$	7.00000i	0.222700i
$989$	−40.0000	−1.27193
$990$	0	0
$991$	13.0000	0.412959	0.206479	−	0.978451i	$-0.433799\pi$
$991$	13.0000	0.412959	0.206479	+	0.978451i	$0.433799\pi$
$992$	4.00000i	0.127000i
$993$	0	0
$994$	−60.0000	−1.90308
$995$	0	0
$996$	0	0
$997$	10.0000i	0.316703i	0.987383	+	0.158352i	$0.0506179\pi$
$997$	10.0000i	0.316703i	−0.987383	+	0.158352i	$0.949382\pi$
$998$	23.0000i	0.728052i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5850.2.e.c.5149.2		2
3.2	odd	2		1950.2.e.n.1249.1		2
5.2	odd	4		5850.2.a.a.1.1		1
5.3	odd	4		5850.2.a.by.1.1		1
5.4	even	2	inner	5850.2.e.c.5149.1		2
15.2	even	4		1950.2.a.x.1.1	yes	1
15.8	even	4		1950.2.a.e.1.1	✓	1
15.14	odd	2		1950.2.e.n.1249.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1950.2.a.e.1.1	✓	1	15.8	even	4
1950.2.a.x.1.1	yes	1	15.2	even	4
1950.2.e.n.1249.1		2	3.2	odd	2
1950.2.e.n.1249.2		2	15.14	odd	2
5850.2.a.a.1.1		1	5.2	odd	4
5850.2.a.by.1.1		1	5.3	odd	4
5850.2.e.c.5149.1		2	5.4	even	2	inner
5850.2.e.c.5149.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5850.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +4.00000i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +4.00000i q^{7} -1.00000i q^{8} -4.00000 q^{11} +1.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} -7.00000 q^{19} -4.00000i q^{22} -4.00000i q^{23} -1.00000 q^{26} -4.00000i q^{28} +5.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} +4.00000 q^{34} -9.00000i q^{37} -7.00000i q^{38} +5.00000 q^{41} -10.0000i q^{43} +4.00000 q^{44} +4.00000 q^{46} +3.00000i q^{47} -9.00000 q^{49} -1.00000i q^{52} -9.00000i q^{53} +4.00000 q^{56} +5.00000i q^{58} -6.00000 q^{59} +4.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +7.00000i q^{67} +4.00000i q^{68} +15.0000 q^{71} +12.0000i q^{73} +9.00000 q^{74} +7.00000 q^{76} -16.0000i q^{77} -7.00000 q^{79} +5.00000i q^{82} -6.00000i q^{83} +10.0000 q^{86} +4.00000i q^{88} +14.0000 q^{89} -4.00000 q^{91} +4.00000i q^{92} -3.00000 q^{94} +16.0000i q^{97} -9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} - 8 q^{11} - 8 q^{14} + 2 q^{16} - 14 q^{19} - 2 q^{26} + 10 q^{29} + 8 q^{31} + 8 q^{34} + 10 q^{41} + 8 q^{44} + 8 q^{46} - 18 q^{49} + 8 q^{56} - 12 q^{59} + 8 q^{61} - 2 q^{64} + 30 q^{71} + 18 q^{74} + 14 q^{76} - 14 q^{79} + 20 q^{86} + 28 q^{89} - 8 q^{91} - 6 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 - 8 * q^11 - 8 * q^14 + 2 * q^16 - 14 * q^19 - 2 * q^26 + 10 * q^29 + 8 * q^31 + 8 * q^34 + 10 * q^41 + 8 * q^44 + 8 * q^46 - 18 * q^49 + 8 * q^56 - 12 * q^59 + 8 * q^61 - 2 * q^64 + 30 * q^71 + 18 * q^74 + 14 * q^76 - 14 * q^79 + 20 * q^86 + 28 * q^89 - 8 * q^91 - 6 * q^94

Introduction

L-functions

Modular forms

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