LMFDB - Embedded newform 4950.2.c.w.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4950,2,Mod(199,4950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4950, 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("4950.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4950.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:	$39.5259490005$
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 1650)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4950.199
Dual form			4950.2.c.w.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +1.00000 q^{11} -4.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} +1.00000 q^{19} -1.00000i q^{22} +9.00000i q^{23} -4.00000 q^{26} -1.00000i q^{28} +6.00000 q^{29} -10.0000 q^{31} -1.00000i q^{32} +3.00000 q^{34} +1.00000i q^{37} -1.00000i q^{38} -9.00000 q^{41} -4.00000i q^{43} -1.00000 q^{44} +9.00000 q^{46} -3.00000i q^{47} +6.00000 q^{49} +4.00000i q^{52} +12.0000i q^{53} -1.00000 q^{56} -6.00000i q^{58} -3.00000 q^{59} -4.00000 q^{61} +10.0000i q^{62} -1.00000 q^{64} +4.00000i q^{67} -3.00000i q^{68} -3.00000 q^{71} -4.00000i q^{73} +1.00000 q^{74} -1.00000 q^{76} +1.00000i q^{77} -11.0000 q^{79} +9.00000i q^{82} -6.00000i q^{83} -4.00000 q^{86} +1.00000i q^{88} +4.00000 q^{91} -9.00000i q^{92} -3.00000 q^{94} +1.00000i q^{97} -6.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} + 2 q^{11} + 2 q^{14} + 2 q^{16} + 2 q^{19} - 8 q^{26} + 12 q^{29} - 20 q^{31} + 6 q^{34} - 18 q^{41} - 2 q^{44} + 18 q^{46} + 12 q^{49} - 2 q^{56} - 6 q^{59} - 8 q^{61} - 2 q^{64} - 6 q^{71} + 2 q^{74} - 2 q^{76} - 22 q^{79} - 8 q^{86} + 8 q^{91} - 6 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^11 + 2 * q^14 + 2 * q^16 + 2 * q^19 - 8 * q^26 + 12 * q^29 - 20 * q^31 + 6 * q^34 - 18 * q^41 - 2 * q^44 + 18 * q^46 + 12 * q^49 - 2 * q^56 - 6 * q^59 - 8 * q^61 - 2 * q^64 - 6 * q^71 + 2 * q^74 - 2 * q^76 - 22 * q^79 - 8 * q^86 + 8 * q^91 - 6 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4950\mathbb{Z}\right)^\times$.

$n$	$551$	$2377$	$4501$
$\chi(n)$	$1$	$-1$	$1$

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
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000i	0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$	3.00000i	0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	− 1.00000i	− 0.213201i
$23$	9.00000i	1.87663i	0.345782	+	0.938315i	$0.387614\pi$
$23$	9.00000i	1.87663i	−0.345782	+	0.938315i	$0.612386\pi$
$24$	0	0
$25$	0	0
$26$	−4.00000	−0.784465
$27$	0	0
$28$	− 1.00000i	− 0.188982i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	3.00000	0.514496
$35$	0	0
$36$	0	0
$37$	1.00000i	0.164399i	0.996616	+	0.0821995i	$0.0261945\pi$
$37$	1.00000i	0.164399i	−0.996616	+	0.0821995i	$0.973806\pi$
$38$	− 1.00000i	− 0.162221i
$39$	0	0
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	9.00000	1.32698
$47$	− 3.00000i	− 0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$	− 3.00000i	− 0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	0	0
$52$	4.00000i	0.554700i
$53$	12.0000i	1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$	12.0000i	1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	− 6.00000i	− 0.787839i
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	10.0000i	1.27000i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	− 3.00000i	− 0.363803i
$69$	0	0
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	0	0
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−1.00000	−0.114708
$77$	1.00000i	0.113961i
$78$	0	0
$79$	−11.0000	−1.23760	−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000	−1.23760	−0.618798	+	0.785550i	$0.712380\pi$
$80$	0	0
$81$	0	0
$82$	9.00000i	0.993884i
$83$	− 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
$86$	−4.00000	−0.431331
$87$	0	0
$88$	1.00000i	0.106600i
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	− 9.00000i	− 0.938315i
$93$	0	0
$94$	−3.00000	−0.309426
$95$	0	0
$96$	0	0
$97$	1.00000i	0.101535i	0.998711	+	0.0507673i	$0.0161667\pi$
$97$	1.00000i	0.101535i	−0.998711	+	0.0507673i	$0.983833\pi$
$98$	− 6.00000i	− 0.606092i
$99$	0	0
$100$	0	0
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	0	0
$103$	− 10.0000i	− 0.985329i	−0.870219	−	0.492665i	$-0.836023\pi$
$103$	− 10.0000i	− 0.985329i	0.870219	−	0.492665i	$-0.163977\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	12.0000	1.16554
$107$	6.00000i	0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$107$	6.00000i	0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$	0	0
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	0	0
$112$	1.00000i	0.0944911i
$113$	12.0000i	1.12887i	0.825479	+	0.564433i	$0.190905\pi$
$113$	12.0000i	1.12887i	−0.825479	+	0.564433i	$0.809095\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	0	0
$118$	3.00000i	0.276172i
$119$	−3.00000	−0.275010
$120$	0	0
$121$	1.00000	0.0909091
$122$	4.00000i	0.362143i
$123$	0	0
$124$	10.0000	0.898027
$125$	0	0
$126$	0	0
$127$	7.00000i	0.621150i	0.950549	+	0.310575i	$0.100522\pi$
$127$	7.00000i	0.621150i	−0.950549	+	0.310575i	$0.899478\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	1.00000i	0.0867110i
$134$	4.00000	0.345547
$135$	0	0
$136$	−3.00000	−0.257248
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	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
$142$	3.00000i	0.251754i
$143$	− 4.00000i	− 0.334497i
$144$	0	0
$145$	0	0
$146$	−4.00000	−0.331042
$147$	0	0
$148$	− 1.00000i	− 0.0821995i
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	1.00000i	0.0811107i
$153$	0	0
$154$	1.00000	0.0805823
$155$	0	0
$156$	0	0
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	11.0000i	0.875113i
$159$	0	0
$160$	0	0
$161$	−9.00000	−0.709299
$162$	0	0
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	−6.00000	−0.465690
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	4.00000i	0.304997i
$173$	15.0000i	1.14043i	0.821496	+	0.570214i	$0.193140\pi$
$173$	15.0000i	1.14043i	−0.821496	+	0.570214i	$0.806860\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	0	0
$179$	−15.0000	−1.12115	−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000	−1.12115	−0.560576	+	0.828103i	$0.689420\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	− 4.00000i	− 0.296500i
$183$	0	0
$184$	−9.00000	−0.663489
$185$	0	0
$186$	0	0
$187$	3.00000i	0.219382i
$188$	3.00000i	0.218797i
$189$	0	0
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	0	0
$193$	20.0000i	1.43963i	0.694165	+	0.719816i	$0.255774\pi$
$193$	20.0000i	1.43963i	−0.694165	+	0.719816i	$0.744226\pi$
$194$	1.00000	0.0717958
$195$	0	0
$196$	−6.00000	−0.428571
$197$	− 3.00000i	− 0.213741i	−0.994273	−	0.106871i	$-0.965917\pi$
$197$	− 3.00000i	− 0.213741i	0.994273	−	0.106871i	$-0.0340831\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	9.00000i	0.633238i
$203$	6.00000i	0.421117i
$204$	0	0
$205$	0	0
$206$	−10.0000	−0.696733
$207$	0	0
$208$	− 4.00000i	− 0.277350i
$209$	1.00000	0.0691714
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	− 12.0000i	− 0.824163i
$213$	0	0
$214$	6.00000	0.410152
$215$	0	0
$216$	0	0
$217$	− 10.0000i	− 0.678844i
$218$	− 16.0000i	− 1.08366i
$219$	0	0
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	20.0000i	1.33930i	0.742677	+	0.669650i	$0.233556\pi$
$223$	20.0000i	1.33930i	−0.742677	+	0.669650i	$0.766444\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	12.0000	0.798228
$227$	18.0000i	1.19470i	0.801980	+	0.597351i	$0.203780\pi$
$227$	18.0000i	1.19470i	−0.801980	+	0.597351i	$0.796220\pi$
$228$	0	0
$229$	−29.0000	−1.91637	−0.958187	−	0.286143i	$-0.907627\pi$
$229$	−29.0000	−1.91637	−0.958187	+	0.286143i	$0.907627\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	15.0000i	0.982683i	0.870967	+	0.491341i	$0.163493\pi$
$233$	15.0000i	0.982683i	−0.870967	+	0.491341i	$0.836507\pi$
$234$	0	0
$235$	0	0
$236$	3.00000	0.195283
$237$	0	0
$238$	3.00000i	0.194461i
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	− 1.00000i	− 0.0642824i
$243$	0	0
$244$	4.00000	0.256074
$245$	0	0
$246$	0	0
$247$	− 4.00000i	− 0.254514i
$248$	− 10.0000i	− 0.635001i
$249$	0	0
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	9.00000i	0.565825i
$254$	7.00000	0.439219
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	0	0
$259$	−1.00000	−0.0621370
$260$	0	0
$261$	0	0
$262$	− 6.00000i	− 0.370681i
$263$	6.00000i	0.369976i	0.982741	+	0.184988i	$0.0592246\pi$
$263$	6.00000i	0.369976i	−0.982741	+	0.184988i	$0.940775\pi$
$264$	0	0
$265$	0	0
$266$	1.00000	0.0613139
$267$	0	0
$268$	− 4.00000i	− 0.244339i
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−25.0000	−1.51864	−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000	−1.51864	−0.759321	+	0.650716i	$0.774469\pi$
$272$	3.00000i	0.181902i
$273$	0	0
$274$	18.0000	1.08742
$275$	0	0
$276$	0	0
$277$	16.0000i	0.961347i	0.876900	+	0.480673i	$0.159608\pi$
$277$	16.0000i	0.961347i	−0.876900	+	0.480673i	$0.840392\pi$
$278$	− 4.00000i	− 0.239904i
$279$	0	0
$280$	0	0
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	0	0
$283$	− 19.0000i	− 1.12943i	−0.825285	−	0.564716i	$-0.808986\pi$
$283$	− 19.0000i	− 1.12943i	0.825285	−	0.564716i	$-0.191014\pi$
$284$	3.00000	0.178017
$285$	0	0
$286$	−4.00000	−0.236525
$287$	− 9.00000i	− 0.531253i
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	4.00000i	0.234082i
$293$	− 3.00000i	− 0.175262i	−0.996153	−	0.0876309i	$-0.972070\pi$
$293$	− 3.00000i	− 0.175262i	0.996153	−	0.0876309i	$-0.0279296\pi$
$294$	0	0
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	0	0
$298$	− 15.0000i	− 0.868927i
$299$	36.0000	2.08193
$300$	0	0
$301$	4.00000	0.230556
$302$	16.0000i	0.920697i
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	− 1.00000i	− 0.0569803i
$309$	0	0
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	− 1.00000i	− 0.0565233i	−0.999601	−	0.0282617i	$-0.991003\pi$
$313$	− 1.00000i	− 0.0565233i	0.999601	−	0.0282617i	$-0.00899717\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	11.0000	0.618798
$317$	12.0000i	0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$	12.0000i	0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	0	0
$322$	9.00000i	0.501550i
$323$	3.00000i	0.166924i
$324$	0	0
$325$	0	0
$326$	20.0000	1.10770
$327$	0	0
$328$	− 9.00000i	− 0.496942i
$329$	3.00000	0.165395
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	6.00000i	0.329293i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 32.0000i	− 1.74315i	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	− 32.0000i	− 1.74315i	0.490261	−	0.871576i	$-0.336901\pi$
$338$	3.00000i	0.163178i
$339$	0	0
$340$	0	0
$341$	−10.0000	−0.541530
$342$	0	0
$343$	13.0000i	0.701934i
$344$	4.00000	0.215666
$345$	0	0
$346$	15.0000	0.806405
$347$	24.0000i	1.28839i	0.764862	+	0.644194i	$0.222807\pi$
$347$	24.0000i	1.28839i	−0.764862	+	0.644194i	$0.777193\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	0	0
$352$	− 1.00000i	− 0.0533002i
$353$	− 12.0000i	− 0.638696i	−0.947638	−	0.319348i	$-0.896536\pi$
$353$	− 12.0000i	− 0.638696i	0.947638	−	0.319348i	$-0.103464\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	15.0000i	0.792775i
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	7.00000i	0.367912i
$363$	0	0
$364$	−4.00000	−0.209657
$365$	0	0
$366$	0	0
$367$	− 26.0000i	− 1.35719i	−0.734513	−	0.678594i	$-0.762589\pi$
$367$	− 26.0000i	− 1.35719i	0.734513	−	0.678594i	$-0.237411\pi$
$368$	9.00000i	0.469157i
$369$	0	0
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	14.0000i	0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$	14.0000i	0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	3.00000	0.155126
$375$	0	0
$376$	3.00000	0.154713
$377$	− 24.0000i	− 1.23606i
$378$	0	0
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	0	0
$381$	0	0
$382$	3.00000i	0.153493i
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	0	0
$385$	0	0
$386$	20.0000	1.01797
$387$	0	0
$388$	− 1.00000i	− 0.0507673i
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	−27.0000	−1.36545
$392$	6.00000i	0.303046i
$393$	0	0
$394$	−3.00000	−0.151138
$395$	0	0
$396$	0	0
$397$	− 38.0000i	− 1.90717i	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	− 38.0000i	− 1.90717i	0.301131	−	0.953583i	$-0.402636\pi$
$398$	− 16.0000i	− 0.802008i
$399$	0	0
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	0	0
$403$	40.0000i	1.99254i
$404$	9.00000	0.447767
$405$	0	0
$406$	6.00000	0.297775
$407$	1.00000i	0.0495682i
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	0	0
$412$	10.0000i	0.492665i
$413$	− 3.00000i	− 0.147620i
$414$	0	0
$415$	0	0
$416$	−4.00000	−0.196116
$417$	0	0
$418$	− 1.00000i	− 0.0489116i
$419$	3.00000	0.146560	0.0732798	−	0.997311i	$-0.476653\pi$
$419$	3.00000	0.146560	0.0732798	+	0.997311i	$0.476653\pi$
$420$	0	0
$421$	35.0000	1.70580	0.852898	−	0.522078i	$-0.174843\pi$
$421$	35.0000	1.70580	0.852898	+	0.522078i	$0.174843\pi$
$422$	4.00000i	0.194717i
$423$	0	0
$424$	−12.0000	−0.582772
$425$	0	0
$426$	0	0
$427$	− 4.00000i	− 0.193574i
$428$	− 6.00000i	− 0.290021i
$429$	0	0
$430$	0	0
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	0	0
$433$	− 22.0000i	− 1.05725i	−0.848855	−	0.528626i	$-0.822707\pi$
$433$	− 22.0000i	− 1.05725i	0.848855	−	0.528626i	$-0.177293\pi$
$434$	−10.0000	−0.480015
$435$	0	0
$436$	−16.0000	−0.766261
$437$	9.00000i	0.430528i
$438$	0	0
$439$	1.00000	0.0477274	0.0238637	−	0.999715i	$-0.492403\pi$
$439$	1.00000	0.0477274	0.0238637	+	0.999715i	$0.492403\pi$
$440$	0	0
$441$	0	0
$442$	− 12.0000i	− 0.570782i
$443$	3.00000i	0.142534i	0.997457	+	0.0712672i	$0.0227043\pi$
$443$	3.00000i	0.142534i	−0.997457	+	0.0712672i	$0.977296\pi$
$444$	0	0
$445$	0	0
$446$	20.0000	0.947027
$447$	0	0
$448$	− 1.00000i	− 0.0472456i
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	−9.00000	−0.423793
$452$	− 12.0000i	− 0.564433i
$453$	0	0
$454$	18.0000	0.844782
$455$	0	0
$456$	0	0
$457$	− 26.0000i	− 1.21623i	−0.793849	−	0.608114i	$-0.791926\pi$
$457$	− 26.0000i	− 1.21623i	0.793849	−	0.608114i	$-0.208074\pi$
$458$	29.0000i	1.35508i
$459$	0	0
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	14.0000i	0.650635i	0.945605	+	0.325318i	$0.105471\pi$
$463$	14.0000i	0.650635i	−0.945605	+	0.325318i	$0.894529\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	15.0000	0.694862
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	0	0
$472$	− 3.00000i	− 0.138086i
$473$	− 4.00000i	− 0.183920i
$474$	0	0
$475$	0	0
$476$	3.00000	0.137505
$477$	0	0
$478$	12.0000i	0.548867i
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	10.0000i	0.455488i
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	0	0
$487$	34.0000i	1.54069i	0.637629	+	0.770344i	$0.279915\pi$
$487$	34.0000i	1.54069i	−0.637629	+	0.770344i	$0.720085\pi$
$488$	− 4.00000i	− 0.181071i
$489$	0	0
$490$	0	0
$491$	−42.0000	−1.89543	−0.947717	−	0.319113i	$-0.896615\pi$
$491$	−42.0000	−1.89543	−0.947717	+	0.319113i	$0.896615\pi$
$492$	0	0
$493$	18.0000i	0.810679i
$494$	−4.00000	−0.179969
$495$	0	0
$496$	−10.0000	−0.449013
$497$	− 3.00000i	− 0.134568i
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	0	0
$502$	− 24.0000i	− 1.07117i
$503$	12.0000i	0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$	12.0000i	0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$	0	0
$505$	0	0
$506$	9.00000	0.400099
$507$	0	0
$508$	− 7.00000i	− 0.310575i
$509$	−12.0000	−0.531891	−0.265945	−	0.963988i	$-0.585684\pi$
$509$	−12.0000	−0.531891	−0.265945	+	0.963988i	$0.585684\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	18.0000	0.793946
$515$	0	0
$516$	0	0
$517$	− 3.00000i	− 0.131940i
$518$	1.00000i	0.0439375i
$519$	0	0
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	35.0000i	1.53044i	0.643767	+	0.765222i	$0.277371\pi$
$523$	35.0000i	1.53044i	−0.643767	+	0.765222i	$0.722629\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	6.00000	0.261612
$527$	− 30.0000i	− 1.30682i
$528$	0	0
$529$	−58.0000	−2.52174
$530$	0	0
$531$	0	0
$532$	− 1.00000i	− 0.0433555i
$533$	36.0000i	1.55933i
$534$	0	0
$535$	0	0
$536$	−4.00000	−0.172774
$537$	0	0
$538$	− 6.00000i	− 0.258678i
$539$	6.00000	0.258438
$540$	0	0
$541$	32.0000	1.37579	0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000	1.37579	0.687894	+	0.725811i	$0.258536\pi$
$542$	25.0000i	1.07384i
$543$	0	0
$544$	3.00000	0.128624
$545$	0	0
$546$	0	0
$547$	37.0000i	1.58201i	0.611812	+	0.791003i	$0.290441\pi$
$547$	37.0000i	1.58201i	−0.611812	+	0.791003i	$0.709559\pi$
$548$	− 18.0000i	− 0.768922i
$549$	0	0
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	− 11.0000i	− 0.467768i
$554$	16.0000	0.679775
$555$	0	0
$556$	−4.00000	−0.169638
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	− 27.0000i	− 1.13893i
$563$	− 30.0000i	− 1.26435i	−0.774826	−	0.632175i	$-0.782163\pi$
$563$	− 30.0000i	− 1.26435i	0.774826	−	0.632175i	$-0.217837\pi$
$564$	0	0
$565$	0	0
$566$	−19.0000	−0.798630
$567$	0	0
$568$	− 3.00000i	− 0.125877i
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	0	0
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	4.00000i	0.167248i
$573$	0	0
$574$	−9.00000	−0.375653
$575$	0	0
$576$	0	0
$577$	19.0000i	0.790980i	0.918470	+	0.395490i	$0.129425\pi$
$577$	19.0000i	0.790980i	−0.918470	+	0.395490i	$0.870575\pi$
$578$	− 8.00000i	− 0.332756i
$579$	0	0
$580$	0	0
$581$	6.00000	0.248922
$582$	0	0
$583$	12.0000i	0.496989i
$584$	4.00000	0.165521
$585$	0	0
$586$	−3.00000	−0.123929
$587$	− 39.0000i	− 1.60970i	−0.593477	−	0.804851i	$-0.702245\pi$
$587$	− 39.0000i	− 1.60970i	0.593477	−	0.804851i	$-0.297755\pi$
$588$	0	0
$589$	−10.0000	−0.412043
$590$	0	0
$591$	0	0
$592$	1.00000i	0.0410997i
$593$	30.0000i	1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$	30.0000i	1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$	0	0
$595$	0	0
$596$	−15.0000	−0.614424
$597$	0	0
$598$	− 36.0000i	− 1.47215i
$599$	−39.0000	−1.59350	−0.796748	−	0.604311i	$-0.793448\pi$
$599$	−39.0000	−1.59350	−0.796748	+	0.604311i	$0.793448\pi$
$600$	0	0
$601$	20.0000	0.815817	0.407909	−	0.913023i	$-0.366258\pi$
$601$	20.0000	0.815817	0.407909	+	0.913023i	$0.366258\pi$
$602$	− 4.00000i	− 0.163028i
$603$	0	0
$604$	16.0000	0.651031
$605$	0	0
$606$	0	0
$607$	− 32.0000i	− 1.29884i	−0.760430	−	0.649420i	$-0.775012\pi$
$607$	− 32.0000i	− 1.29884i	0.760430	−	0.649420i	$-0.224988\pi$
$608$	− 1.00000i	− 0.0405554i
$609$	0	0
$610$	0	0
$611$	−12.0000	−0.485468
$612$	0	0
$613$	− 4.00000i	− 0.161558i	−0.996732	−	0.0807792i	$-0.974259\pi$
$613$	− 4.00000i	− 0.161558i	0.996732	−	0.0807792i	$-0.0257409\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	0	0
$619$	34.0000	1.36658	0.683288	−	0.730149i	$-0.260549\pi$
$619$	34.0000	1.36658	0.683288	+	0.730149i	$0.260549\pi$
$620$	0	0
$621$	0	0
$622$	24.0000i	0.962312i
$623$	0	0
$624$	0	0
$625$	0	0
$626$	−1.00000	−0.0399680
$627$	0	0
$628$	− 10.0000i	− 0.399043i
$629$	−3.00000	−0.119618
$630$	0	0
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	− 11.0000i	− 0.437557i
$633$	0	0
$634$	12.0000	0.476581
$635$	0	0
$636$	0	0
$637$	− 24.0000i	− 0.950915i
$638$	− 6.00000i	− 0.237542i
$639$	0	0
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	− 22.0000i	− 0.867595i	−0.901010	−	0.433798i	$-0.857173\pi$
$643$	− 22.0000i	− 0.867595i	0.901010	−	0.433798i	$-0.142827\pi$
$644$	9.00000	0.354650
$645$	0	0
$646$	3.00000	0.118033
$647$	33.0000i	1.29736i	0.761060	+	0.648682i	$0.224679\pi$
$647$	33.0000i	1.29736i	−0.761060	+	0.648682i	$0.775321\pi$
$648$	0	0
$649$	−3.00000	−0.117760
$650$	0	0
$651$	0	0
$652$	− 20.0000i	− 0.783260i
$653$	18.0000i	0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$	18.0000i	0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$	0	0
$655$	0	0
$656$	−9.00000	−0.351391
$657$	0	0
$658$	− 3.00000i	− 0.116952i
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	17.0000	0.661223	0.330612	−	0.943767i	$-0.392745\pi$
$661$	17.0000	0.661223	0.330612	+	0.943767i	$0.392745\pi$
$662$	− 20.0000i	− 0.777322i
$663$	0	0
$664$	6.00000	0.232845
$665$	0	0
$666$	0	0
$667$	54.0000i	2.09089i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	− 16.0000i	− 0.616755i	−0.951264	−	0.308377i	$-0.900214\pi$
$673$	− 16.0000i	− 0.616755i	0.951264	−	0.308377i	$-0.0997859\pi$
$674$	−32.0000	−1.23259
$675$	0	0
$676$	3.00000	0.115385
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	0	0
$679$	−1.00000	−0.0383765
$680$	0	0
$681$	0	0
$682$	10.0000i	0.382920i
$683$	− 3.00000i	− 0.114792i	−0.998351	−	0.0573959i	$-0.981720\pi$
$683$	− 3.00000i	− 0.114792i	0.998351	−	0.0573959i	$-0.0182797\pi$
$684$	0	0
$685$	0	0
$686$	13.0000	0.496342
$687$	0	0
$688$	− 4.00000i	− 0.152499i
$689$	48.0000	1.82865
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	− 15.0000i	− 0.570214i
$693$	0	0
$694$	24.0000	0.911028
$695$	0	0
$696$	0	0
$697$	− 27.0000i	− 1.02270i
$698$	14.0000i	0.529908i
$699$	0	0
$700$	0	0
$701$	−15.0000	−0.566542	−0.283271	−	0.959040i	$-0.591420\pi$
$701$	−15.0000	−0.566542	−0.283271	+	0.959040i	$0.591420\pi$
$702$	0	0
$703$	1.00000i	0.0377157i
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−12.0000	−0.451626
$707$	− 9.00000i	− 0.338480i
$708$	0	0
$709$	43.0000	1.61490	0.807449	−	0.589937i	$-0.200847\pi$
$709$	43.0000	1.61490	0.807449	+	0.589937i	$0.200847\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 90.0000i	− 3.37053i
$714$	0	0
$715$	0	0
$716$	15.0000	0.560576
$717$	0	0
$718$	− 30.0000i	− 1.11959i
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	10.0000	0.372419
$722$	18.0000i	0.669891i
$723$	0	0
$724$	7.00000	0.260153
$725$	0	0
$726$	0	0
$727$	40.0000i	1.48352i	0.670667	+	0.741759i	$0.266008\pi$
$727$	40.0000i	1.48352i	−0.670667	+	0.741759i	$0.733992\pi$
$728$	4.00000i	0.148250i
$729$	0	0
$730$	0	0
$731$	12.0000	0.443836
$732$	0	0
$733$	32.0000i	1.18195i	0.806691	+	0.590973i	$0.201256\pi$
$733$	32.0000i	1.18195i	−0.806691	+	0.590973i	$0.798744\pi$
$734$	−26.0000	−0.959678
$735$	0	0
$736$	9.00000	0.331744
$737$	4.00000i	0.147342i
$738$	0	0
$739$	−29.0000	−1.06678	−0.533391	−	0.845869i	$-0.679083\pi$
$739$	−29.0000	−1.06678	−0.533391	+	0.845869i	$0.679083\pi$
$740$	0	0
$741$	0	0
$742$	12.0000i	0.440534i
$743$	− 12.0000i	− 0.440237i	−0.975473	−	0.220119i	$-0.929356\pi$
$743$	− 12.0000i	− 0.440237i	0.975473	−	0.220119i	$-0.0706445\pi$
$744$	0	0
$745$	0	0
$746$	14.0000	0.512576
$747$	0	0
$748$	− 3.00000i	− 0.109691i
$749$	−6.00000	−0.219235
$750$	0	0
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	− 3.00000i	− 0.109399i
$753$	0	0
$754$	−24.0000	−0.874028
$755$	0	0
$756$	0	0
$757$	− 2.00000i	− 0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$	− 2.00000i	− 0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$	2.00000i	0.0726433i
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	16.0000i	0.579239i
$764$	3.00000	0.108536
$765$	0	0
$766$	−24.0000	−0.867155
$767$	12.0000i	0.433295i
$768$	0	0
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	0	0
$772$	− 20.0000i	− 0.719816i
$773$	48.0000i	1.72644i	0.504828	+	0.863220i	$0.331556\pi$
$773$	48.0000i	1.72644i	−0.504828	+	0.863220i	$0.668444\pi$
$774$	0	0
$775$	0	0
$776$	−1.00000	−0.0358979
$777$	0	0
$778$	− 30.0000i	− 1.07555i
$779$	−9.00000	−0.322458
$780$	0	0
$781$	−3.00000	−0.107348
$782$	27.0000i	0.965518i
$783$	0	0
$784$	6.00000	0.214286
$785$	0	0
$786$	0	0
$787$	− 35.0000i	− 1.24762i	−0.781578	−	0.623808i	$-0.785585\pi$
$787$	− 35.0000i	− 1.24762i	0.781578	−	0.623808i	$-0.214415\pi$
$788$	3.00000i	0.106871i
$789$	0	0
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	16.0000i	0.568177i
$794$	−38.0000	−1.34857
$795$	0	0
$796$	−16.0000	−0.567105
$797$	− 6.00000i	− 0.212531i	−0.994338	−	0.106265i	$-0.966111\pi$
$797$	− 6.00000i	− 0.212531i	0.994338	−	0.106265i	$-0.0338893\pi$
$798$	0	0
$799$	9.00000	0.318397
$800$	0	0
$801$	0	0
$802$	12.0000i	0.423735i
$803$	− 4.00000i	− 0.141157i
$804$	0	0
$805$	0	0
$806$	40.0000	1.40894
$807$	0	0
$808$	− 9.00000i	− 0.316619i
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	0	0
$811$	47.0000	1.65039	0.825197	−	0.564846i	$-0.191064\pi$
$811$	47.0000	1.65039	0.825197	+	0.564846i	$0.191064\pi$
$812$	− 6.00000i	− 0.210559i
$813$	0	0
$814$	1.00000	0.0350500
$815$	0	0
$816$	0	0
$817$	− 4.00000i	− 0.139942i
$818$	− 4.00000i	− 0.139857i
$819$	0	0
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	0	0
$823$	− 40.0000i	− 1.39431i	−0.716919	−	0.697156i	$-0.754448\pi$
$823$	− 40.0000i	− 1.39431i	0.716919	−	0.697156i	$-0.245552\pi$
$824$	10.0000	0.348367
$825$	0	0
$826$	−3.00000	−0.104383
$827$	− 48.0000i	− 1.66912i	−0.550914	−	0.834562i	$-0.685721\pi$
$827$	− 48.0000i	− 1.66912i	0.550914	−	0.834562i	$-0.314279\pi$
$828$	0	0
$829$	−26.0000	−0.903017	−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000	−0.903017	−0.451509	+	0.892267i	$0.649114\pi$
$830$	0	0
$831$	0	0
$832$	4.00000i	0.138675i
$833$	18.0000i	0.623663i
$834$	0	0
$835$	0	0
$836$	−1.00000	−0.0345857
$837$	0	0
$838$	− 3.00000i	− 0.103633i
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 35.0000i	− 1.20618i
$843$	0	0
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	1.00000i	0.0343604i
$848$	12.0000i	0.412082i
$849$	0	0
$850$	0	0
$851$	−9.00000	−0.308516
$852$	0	0
$853$	26.0000i	0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$	26.0000i	0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	−6.00000	−0.205076
$857$	− 21.0000i	− 0.717346i	−0.933463	−	0.358673i	$-0.883229\pi$
$857$	− 21.0000i	− 0.717346i	0.933463	−	0.358673i	$-0.116771\pi$
$858$	0	0
$859$	16.0000	0.545913	0.272956	−	0.962026i	$-0.411998\pi$
$859$	16.0000	0.545913	0.272956	+	0.962026i	$0.411998\pi$
$860$	0	0
$861$	0	0
$862$	− 30.0000i	− 1.02180i
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	−22.0000	−0.747590
$867$	0	0
$868$	10.0000i	0.339422i
$869$	−11.0000	−0.373149
$870$	0	0
$871$	16.0000	0.542139
$872$	16.0000i	0.541828i
$873$	0	0
$874$	9.00000	0.304430
$875$	0	0
$876$	0	0
$877$	− 38.0000i	− 1.28317i	−0.767052	−	0.641584i	$-0.778277\pi$
$877$	− 38.0000i	− 1.28317i	0.767052	−	0.641584i	$-0.221723\pi$
$878$	− 1.00000i	− 0.0337484i
$879$	0	0
$880$	0	0
$881$	−36.0000	−1.21287	−0.606435	−	0.795133i	$-0.707401\pi$
$881$	−36.0000	−1.21287	−0.606435	+	0.795133i	$0.707401\pi$
$882$	0	0
$883$	− 34.0000i	− 1.14419i	−0.820187	−	0.572096i	$-0.806131\pi$
$883$	− 34.0000i	− 1.14419i	0.820187	−	0.572096i	$-0.193869\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	3.00000	0.100787
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−7.00000	−0.234772
$890$	0	0
$891$	0	0
$892$	− 20.0000i	− 0.669650i
$893$	− 3.00000i	− 0.100391i
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	12.0000i	0.400445i
$899$	−60.0000	−2.00111
$900$	0	0
$901$	−36.0000	−1.19933
$902$	9.00000i	0.299667i
$903$	0	0
$904$	−12.0000	−0.399114
$905$	0	0
$906$	0	0
$907$	58.0000i	1.92586i	0.269754	+	0.962929i	$0.413058\pi$
$907$	58.0000i	1.92586i	−0.269754	+	0.962929i	$0.586942\pi$
$908$	− 18.0000i	− 0.597351i
$909$	0	0
$910$	0	0
$911$	−27.0000	−0.894550	−0.447275	−	0.894397i	$-0.647605\pi$
$911$	−27.0000	−0.894550	−0.447275	+	0.894397i	$0.647605\pi$
$912$	0	0
$913$	− 6.00000i	− 0.198571i
$914$	−26.0000	−0.860004
$915$	0	0
$916$	29.0000	0.958187
$917$	6.00000i	0.198137i
$918$	0	0
$919$	−11.0000	−0.362857	−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000	−0.362857	−0.181428	+	0.983404i	$0.558072\pi$
$920$	0	0
$921$	0	0
$922$	− 6.00000i	− 0.197599i
$923$	12.0000i	0.394985i
$924$	0	0
$925$	0	0
$926$	14.0000	0.460069
$927$	0	0
$928$	− 6.00000i	− 0.196960i
$929$	12.0000	0.393707	0.196854	−	0.980433i	$-0.436928\pi$
$929$	12.0000	0.393707	0.196854	+	0.980433i	$0.436928\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	− 15.0000i	− 0.491341i
$933$	0	0
$934$	−12.0000	−0.392652
$935$	0	0
$936$	0	0
$937$	− 38.0000i	− 1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	− 38.0000i	− 1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$	4.00000i	0.130605i
$939$	0	0
$940$	0	0
$941$	27.0000	0.880175	0.440087	−	0.897955i	$-0.354947\pi$
$941$	27.0000	0.880175	0.440087	+	0.897955i	$0.354947\pi$
$942$	0	0
$943$	− 81.0000i	− 2.63772i
$944$	−3.00000	−0.0976417
$945$	0	0
$946$	−4.00000	−0.130051
$947$	− 27.0000i	− 0.877382i	−0.898638	−	0.438691i	$-0.855442\pi$
$947$	− 27.0000i	− 0.877382i	0.898638	−	0.438691i	$-0.144558\pi$
$948$	0	0
$949$	−16.0000	−0.519382
$950$	0	0
$951$	0	0
$952$	− 3.00000i	− 0.0972306i
$953$	− 51.0000i	− 1.65205i	−0.563632	−	0.826026i	$-0.690596\pi$
$953$	− 51.0000i	− 1.65205i	0.563632	−	0.826026i	$-0.309404\pi$
$954$	0	0
$955$	0	0
$956$	12.0000	0.388108
$957$	0	0
$958$	0	0
$959$	−18.0000	−0.581250
$960$	0	0
$961$	69.0000	2.22581
$962$	− 4.00000i	− 0.128965i
$963$	0	0
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	16.0000i	0.514525i	0.966342	+	0.257263i	$0.0828206\pi$
$967$	16.0000i	0.514525i	−0.966342	+	0.257263i	$0.917179\pi$
$968$	1.00000i	0.0321412i
$969$	0	0
$970$	0	0
$971$	−27.0000	−0.866471	−0.433236	−	0.901281i	$-0.642628\pi$
$971$	−27.0000	−0.866471	−0.433236	+	0.901281i	$0.642628\pi$
$972$	0	0
$973$	4.00000i	0.128234i
$974$	34.0000	1.08943
$975$	0	0
$976$	−4.00000	−0.128037
$977$	− 6.00000i	− 0.191957i	−0.995383	−	0.0959785i	$-0.969402\pi$
$977$	− 6.00000i	− 0.191957i	0.995383	−	0.0959785i	$-0.0305980\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	42.0000i	1.34027i
$983$	− 9.00000i	− 0.287055i	−0.989646	−	0.143528i	$-0.954155\pi$
$983$	− 9.00000i	− 0.287055i	0.989646	−	0.143528i	$-0.0458446\pi$
$984$	0	0
$985$	0	0
$986$	18.0000	0.573237
$987$	0	0
$988$	4.00000i	0.127257i
$989$	36.0000	1.14473
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	10.0000i	0.317500i
$993$	0	0
$994$	−3.00000	−0.0951542
$995$	0	0
$996$	0	0
$997$	46.0000i	1.45683i	0.685134	+	0.728417i	$0.259744\pi$
$997$	46.0000i	1.45683i	−0.685134	+	0.728417i	$0.740256\pi$
$998$	− 4.00000i	− 0.126618i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4950.2.c.w.199.1		2
3.2	odd	2		1650.2.c.b.199.2		2
5.2	odd	4		4950.2.a.bf.1.1		1
5.3	odd	4		4950.2.a.o.1.1		1
5.4	even	2	inner	4950.2.c.w.199.2		2
15.2	even	4		1650.2.a.g.1.1	✓	1
15.8	even	4		1650.2.a.o.1.1	yes	1
15.14	odd	2		1650.2.c.b.199.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1650.2.a.g.1.1	✓	1	15.2	even	4
1650.2.a.o.1.1	yes	1	15.8	even	4
1650.2.c.b.199.1		2	15.14	odd	2
1650.2.c.b.199.2		2	3.2	odd	2
4950.2.a.o.1.1		1	5.3	odd	4
4950.2.a.bf.1.1		1	5.2	odd	4
4950.2.c.w.199.1		2	1.1	even	1	trivial
4950.2.c.w.199.2		2	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:	\( N \)	\(=\)	\( 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4950.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +1.00000 q^{11} -4.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} +1.00000 q^{19} -1.00000i q^{22} +9.00000i q^{23} -4.00000 q^{26} -1.00000i q^{28} +6.00000 q^{29} -10.0000 q^{31} -1.00000i q^{32} +3.00000 q^{34} +1.00000i q^{37} -1.00000i q^{38} -9.00000 q^{41} -4.00000i q^{43} -1.00000 q^{44} +9.00000 q^{46} -3.00000i q^{47} +6.00000 q^{49} +4.00000i q^{52} +12.0000i q^{53} -1.00000 q^{56} -6.00000i q^{58} -3.00000 q^{59} -4.00000 q^{61} +10.0000i q^{62} -1.00000 q^{64} +4.00000i q^{67} -3.00000i q^{68} -3.00000 q^{71} -4.00000i q^{73} +1.00000 q^{74} -1.00000 q^{76} +1.00000i q^{77} -11.0000 q^{79} +9.00000i q^{82} -6.00000i q^{83} -4.00000 q^{86} +1.00000i q^{88} +4.00000 q^{91} -9.00000i q^{92} -3.00000 q^{94} +1.00000i q^{97} -6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 2 q^{11} + 2 q^{14} + 2 q^{16} + 2 q^{19} - 8 q^{26} + 12 q^{29} - 20 q^{31} + 6 q^{34} - 18 q^{41} - 2 q^{44} + 18 q^{46} + 12 q^{49} - 2 q^{56} - 6 q^{59} - 8 q^{61} - 2 q^{64} - 6 q^{71} + 2 q^{74} - 2 q^{76} - 22 q^{79} - 8 q^{86} + 8 q^{91} - 6 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^11 + 2 * q^14 + 2 * q^16 + 2 * q^19 - 8 * q^26 + 12 * q^29 - 20 * q^31 + 6 * q^34 - 18 * q^41 - 2 * q^44 + 18 * q^46 + 12 * q^49 - 2 * q^56 - 6 * q^59 - 8 * q^61 - 2 * q^64 - 6 * q^71 + 2 * q^74 - 2 * q^76 - 22 * q^79 - 8 * q^86 + 8 * q^91 - 6 * q^94

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