LMFDB - Embedded newform 3150.2.g.n.2899.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,2,Mod(2899,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, 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("3150.2899");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3150.g (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:	$25.1528766367$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2899.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3150.2899
Dual form			3150.2.g.n.2899.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +1.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} -2.00000 q^{19} +3.00000i q^{23} +1.00000 q^{26} -1.00000i q^{28} +3.00000 q^{29} -1.00000 q^{31} -1.00000i q^{32} -3.00000 q^{34} +2.00000i q^{37} +2.00000i q^{38} +3.00000 q^{41} +7.00000i q^{43} +3.00000 q^{46} -6.00000i q^{47} -1.00000 q^{49} -1.00000i q^{52} +9.00000i q^{53} -1.00000 q^{56} -3.00000i q^{58} +3.00000 q^{59} -1.00000 q^{61} +1.00000i q^{62} -1.00000 q^{64} +8.00000i q^{67} +3.00000i q^{68} +4.00000i q^{73} +2.00000 q^{74} +2.00000 q^{76} -8.00000 q^{79} -3.00000i q^{82} +15.0000i q^{83} +7.00000 q^{86} +6.00000 q^{89} -1.00000 q^{91} -3.00000i q^{92} -6.00000 q^{94} +8.00000i q^{97} +1.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} + 2 q^{14} + 2 q^{16} - 4 q^{19} + 2 q^{26} + 6 q^{29} - 2 q^{31} - 6 q^{34} + 6 q^{41} + 6 q^{46} - 2 q^{49} - 2 q^{56} + 6 q^{59} - 2 q^{61} - 2 q^{64} + 4 q^{74} + 4 q^{76} - 16 q^{79} + 14 q^{86} + 12 q^{89} - 2 q^{91} - 12 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^14 + 2 * q^16 - 4 * q^19 + 2 * q^26 + 6 * q^29 - 2 * q^31 - 6 * q^34 + 6 * q^41 + 6 * q^46 - 2 * q^49 - 2 * q^56 + 6 * q^59 - 2 * q^61 - 2 * q^64 + 4 * q^74 + 4 * q^76 - 16 * q^79 + 14 * q^86 + 12 * q^89 - 2 * q^91 - 12 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$1$	$1$

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
$7$	1.00000i	0.377964i
$8$	1.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.00000i	0.625543i	0.949828	+	0.312772i	$0.101257\pi$
$23$	3.00000i	0.625543i	−0.949828	+	0.312772i	$0.898743\pi$
$24$	0	0
$25$	0	0
$26$	1.00000	0.196116
$27$	0	0
$28$	− 1.00000i	− 0.188982i
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	−3.00000	−0.514496
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	2.00000i	0.324443i
$39$	0	0
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	7.00000i	1.06749i	0.845645	+	0.533745i	$0.179216\pi$
$43$	7.00000i	1.06749i	−0.845645	+	0.533745i	$0.820784\pi$
$44$	0	0
$45$	0	0
$46$	3.00000	0.442326
$47$	− 6.00000i	− 0.875190i	−0.899172	−	0.437595i	$-0.855830\pi$
$47$	− 6.00000i	− 0.875190i	0.899172	−	0.437595i	$-0.144170\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 1.00000i	− 0.138675i
$53$	9.00000i	1.23625i	0.786082	+	0.618123i	$0.212106\pi$
$53$	9.00000i	1.23625i	−0.786082	+	0.618123i	$0.787894\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	− 3.00000i	− 0.393919i
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	1.00000i	0.127000i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	3.00000i	0.363803i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	0	0
$82$	− 3.00000i	− 0.331295i
$83$	15.0000i	1.64646i	0.567705	+	0.823232i	$0.307831\pi$
$83$	15.0000i	1.64646i	−0.567705	+	0.823232i	$0.692169\pi$
$84$	0	0
$85$	0	0
$86$	7.00000	0.754829
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	− 3.00000i	− 0.312772i
$93$	0	0
$94$	−6.00000	−0.618853
$95$	0	0
$96$	0	0
$97$	8.00000i	0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$	8.00000i	0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$	1.00000i	0.101015i
$99$	0	0
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	13.0000i	1.28093i	0.767988	+	0.640464i	$0.221258\pi$
$103$	13.0000i	1.28093i	−0.767988	+	0.640464i	$0.778742\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	9.00000	0.874157
$107$	− 6.00000i	− 0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	− 6.00000i	− 0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	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
$116$	−3.00000	−0.278543
$117$	0	0
$118$	− 3.00000i	− 0.276172i
$119$	3.00000	0.275010
$120$	0	0
$121$	−11.0000	−1.00000
$122$	1.00000i	0.0905357i
$123$	0	0
$124$	1.00000	0.0898027
$125$	0	0
$126$	0	0
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	− 2.00000i	− 0.173422i
$134$	8.00000	0.691095
$135$	0	0
$136$	3.00000	0.257248
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	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
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	4.00000	0.331042
$147$	0	0
$148$	− 2.00000i	− 0.164399i
$149$	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
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	− 2.00000i	− 0.162221i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	8.00000i	0.636446i
$159$	0	0
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	1.00000i	0.0783260i	0.999233	+	0.0391630i	$0.0124692\pi$
$163$	1.00000i	0.0783260i	−0.999233	+	0.0391630i	$0.987531\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	15.0000	1.16423
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	− 7.00000i	− 0.533745i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	− 6.00000i	− 0.449719i
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	1.00000i	0.0741249i
$183$	0	0
$184$	−3.00000	−0.221163
$185$	0	0
$186$	0	0
$187$	0	0
$188$	6.00000i	0.437595i
$189$	0	0
$190$	0	0
$191$	9.00000	0.651217	0.325609	−	0.945505i	$-0.394431\pi$
$191$	9.00000	0.651217	0.325609	+	0.945505i	$0.394431\pi$
$192$	0	0
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	1.00000	0.0714286
$197$	3.00000i	0.213741i	0.994273	+	0.106871i	$0.0340831\pi$
$197$	3.00000i	0.213741i	−0.994273	+	0.106871i	$0.965917\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	0	0
$202$	− 18.0000i	− 1.26648i
$203$	3.00000i	0.210559i
$204$	0	0
$205$	0	0
$206$	13.0000	0.905753
$207$	0	0
$208$	1.00000i	0.0693375i
$209$	0	0
$210$	0	0
$211$	−7.00000	−0.481900	−0.240950	−	0.970538i	$-0.577459\pi$
$211$	−7.00000	−0.481900	−0.240950	+	0.970538i	$0.577459\pi$
$212$	− 9.00000i	− 0.618123i
$213$	0	0
$214$	−6.00000	−0.410152
$215$	0	0
$216$	0	0
$217$	− 1.00000i	− 0.0678844i
$218$	2.00000i	0.135457i
$219$	0	0
$220$	0	0
$221$	3.00000	0.201802
$222$	0	0
$223$	13.0000i	0.870544i	0.900299	+	0.435272i	$0.143348\pi$
$223$	13.0000i	0.870544i	−0.900299	+	0.435272i	$0.856652\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	12.0000	0.798228
$227$	− 21.0000i	− 1.39382i	−0.717159	−	0.696909i	$-0.754558\pi$
$227$	− 21.0000i	− 1.39382i	0.717159	−	0.696909i	$-0.245442\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	0	0
$232$	3.00000i	0.196960i
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	0	0
$235$	0	0
$236$	−3.00000	−0.195283
$237$	0	0
$238$	− 3.00000i	− 0.194461i
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	11.0000i	0.707107i
$243$	0	0
$244$	1.00000	0.0640184
$245$	0	0
$246$	0	0
$247$	− 2.00000i	− 0.127257i
$248$	− 1.00000i	− 0.0635001i
$249$	0	0
$250$	0	0
$251$	−3.00000	−0.189358	−0.0946792	−	0.995508i	$-0.530183\pi$
$251$	−3.00000	−0.189358	−0.0946792	+	0.995508i	$0.530183\pi$
$252$	0	0
$253$	0	0
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 3.00000i	− 0.187135i	−0.995613	−	0.0935674i	$-0.970173\pi$
$257$	− 3.00000i	− 0.187135i	0.995613	−	0.0935674i	$-0.0298271\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	0	0
$261$	0	0
$262$	− 12.0000i	− 0.741362i
$263$	15.0000i	0.924940i	0.886635	+	0.462470i	$0.153037\pi$
$263$	15.0000i	0.924940i	−0.886635	+	0.462470i	$0.846963\pi$
$264$	0	0
$265$	0	0
$266$	−2.00000	−0.122628
$267$	0	0
$268$	− 8.00000i	− 0.488678i
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	− 3.00000i	− 0.181902i
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8.00000i	0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$	8.00000i	0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$	− 4.00000i	− 0.239904i
$279$	0	0
$280$	0	0
$281$	−24.0000	−1.43172	−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000	−1.43172	−0.715860	+	0.698244i	$0.753965\pi$
$282$	0	0
$283$	− 14.0000i	− 0.832214i	−0.909316	−	0.416107i	$-0.863394\pi$
$283$	− 14.0000i	− 0.832214i	0.909316	−	0.416107i	$-0.136606\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.00000i	0.177084i
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	− 4.00000i	− 0.234082i
$293$	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
$296$	−2.00000	−0.116248
$297$	0	0
$298$	− 15.0000i	− 0.868927i
$299$	−3.00000	−0.173494
$300$	0	0
$301$	−7.00000	−0.403473
$302$	10.0000i	0.575435i
$303$	0	0
$304$	−2.00000	−0.114708
$305$	0	0
$306$	0	0
$307$	14.0000i	0.799022i	0.916728	+	0.399511i	$0.130820\pi$
$307$	14.0000i	0.799022i	−0.916728	+	0.399511i	$0.869180\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	0	0
$313$	28.0000i	1.58265i	0.611393	+	0.791327i	$0.290609\pi$
$313$	28.0000i	1.58265i	−0.611393	+	0.791327i	$0.709391\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	8.00000	0.450035
$317$	− 21.0000i	− 1.17948i	−0.807594	−	0.589739i	$-0.799231\pi$
$317$	− 21.0000i	− 1.17948i	0.807594	−	0.589739i	$-0.200769\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	3.00000i	0.167183i
$323$	6.00000i	0.333849i
$324$	0	0
$325$	0	0
$326$	1.00000	0.0553849
$327$	0	0
$328$	3.00000i	0.165647i
$329$	6.00000	0.330791
$330$	0	0
$331$	−25.0000	−1.37412	−0.687062	−	0.726599i	$-0.741100\pi$
$331$	−25.0000	−1.37412	−0.687062	+	0.726599i	$0.741100\pi$
$332$	− 15.0000i	− 0.823232i
$333$	0	0
$334$	12.0000	0.656611
$335$	0	0
$336$	0	0
$337$	− 13.0000i	− 0.708155i	−0.935216	−	0.354078i	$-0.884795\pi$
$337$	− 13.0000i	− 0.708155i	0.935216	−	0.354078i	$-0.115205\pi$
$338$	− 12.0000i	− 0.652714i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	−7.00000	−0.377415
$345$	0	0
$346$	6.00000	0.322562
$347$	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
$349$	31.0000	1.65939	0.829696	−	0.558216i	$-0.188514\pi$
$349$	31.0000	1.65939	0.829696	+	0.558216i	$0.188514\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 18.0000i	− 0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$	− 18.0000i	− 0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	− 6.00000i	− 0.317110i
$359$	−9.00000	−0.475002	−0.237501	−	0.971387i	$-0.576328\pi$
$359$	−9.00000	−0.475002	−0.237501	+	0.971387i	$0.576328\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	− 2.00000i	− 0.105118i
$363$	0	0
$364$	1.00000	0.0524142
$365$	0	0
$366$	0	0
$367$	17.0000i	0.887393i	0.896177	+	0.443696i	$0.146333\pi$
$367$	17.0000i	0.887393i	−0.896177	+	0.443696i	$0.853667\pi$
$368$	3.00000i	0.156386i
$369$	0	0
$370$	0	0
$371$	−9.00000	−0.467257
$372$	0	0
$373$	− 14.0000i	− 0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	− 14.0000i	− 0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$	0	0
$375$	0	0
$376$	6.00000	0.309426
$377$	3.00000i	0.154508i
$378$	0	0
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	0	0
$381$	0	0
$382$	− 9.00000i	− 0.460480i
$383$	− 12.0000i	− 0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$	− 12.0000i	− 0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	0	0
$385$	0	0
$386$	−14.0000	−0.712581
$387$	0	0
$388$	− 8.00000i	− 0.406138i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	9.00000	0.455150
$392$	− 1.00000i	− 0.0505076i
$393$	0	0
$394$	3.00000	0.151138
$395$	0	0
$396$	0	0
$397$	− 31.0000i	− 1.55585i	−0.628360	−	0.777923i	$-0.716273\pi$
$397$	− 31.0000i	− 1.55585i	0.628360	−	0.777923i	$-0.283727\pi$
$398$	8.00000i	0.401004i
$399$	0	0
$400$	0	0
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	− 1.00000i	− 0.0498135i
$404$	−18.0000	−0.895533
$405$	0	0
$406$	3.00000	0.148888
$407$	0	0
$408$	0	0
$409$	28.0000	1.38451	0.692255	−	0.721653i	$-0.256617\pi$
$409$	28.0000	1.38451	0.692255	+	0.721653i	$0.256617\pi$
$410$	0	0
$411$	0	0
$412$	− 13.0000i	− 0.640464i
$413$	3.00000i	0.147620i
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	0	0
$419$	−15.0000	−0.732798	−0.366399	−	0.930458i	$-0.619409\pi$
$419$	−15.0000	−0.732798	−0.366399	+	0.930458i	$0.619409\pi$
$420$	0	0
$421$	−16.0000	−0.779792	−0.389896	−	0.920859i	$-0.627489\pi$
$421$	−16.0000	−0.779792	−0.389896	+	0.920859i	$0.627489\pi$
$422$	7.00000i	0.340755i
$423$	0	0
$424$	−9.00000	−0.437079
$425$	0	0
$426$	0	0
$427$	− 1.00000i	− 0.0483934i
$428$	6.00000i	0.290021i
$429$	0	0
$430$	0	0
$431$	−39.0000	−1.87856	−0.939282	−	0.343146i	$-0.888507\pi$
$431$	−39.0000	−1.87856	−0.939282	+	0.343146i	$0.888507\pi$
$432$	0	0
$433$	− 14.0000i	− 0.672797i	−0.941720	−	0.336399i	$-0.890791\pi$
$433$	− 14.0000i	− 0.672797i	0.941720	−	0.336399i	$-0.109209\pi$
$434$	−1.00000	−0.0480015
$435$	0	0
$436$	2.00000	0.0957826
$437$	− 6.00000i	− 0.287019i
$438$	0	0
$439$	13.0000	0.620456	0.310228	−	0.950662i	$-0.399595\pi$
$439$	13.0000	0.620456	0.310228	+	0.950662i	$0.399595\pi$
$440$	0	0
$441$	0	0
$442$	− 3.00000i	− 0.142695i
$443$	− 18.0000i	− 0.855206i	−0.903967	−	0.427603i	$-0.859358\pi$
$443$	− 18.0000i	− 0.855206i	0.903967	−	0.427603i	$-0.140642\pi$
$444$	0	0
$445$	0	0
$446$	13.0000	0.615568
$447$	0	0
$448$	− 1.00000i	− 0.0472456i
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	0	0
$452$	− 12.0000i	− 0.564433i
$453$	0	0
$454$	−21.0000	−0.985579
$455$	0	0
$456$	0	0
$457$	− 1.00000i	− 0.0467780i	−0.999726	−	0.0233890i	$-0.992554\pi$
$457$	− 1.00000i	− 0.0467780i	0.999726	−	0.0233890i	$-0.00744563\pi$
$458$	− 22.0000i	− 1.02799i
$459$	0	0
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	28.0000i	1.30127i	0.759390	+	0.650635i	$0.225497\pi$
$463$	28.0000i	1.30127i	−0.759390	+	0.650635i	$0.774503\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−6.00000	−0.277945
$467$	− 3.00000i	− 0.138823i	−0.997588	−	0.0694117i	$-0.977888\pi$
$467$	− 3.00000i	− 0.138823i	0.997588	−	0.0694117i	$-0.0221122\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	0	0
$472$	3.00000i	0.138086i
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−3.00000	−0.137505
$477$	0	0
$478$	− 24.0000i	− 1.09773i
$479$	−30.0000	−1.37073	−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000	−1.37073	−0.685367	+	0.728197i	$0.740358\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	22.0000i	1.00207i
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	0	0
$487$	20.0000i	0.906287i	0.891438	+	0.453143i	$0.149697\pi$
$487$	20.0000i	0.906287i	−0.891438	+	0.453143i	$0.850303\pi$
$488$	− 1.00000i	− 0.0452679i
$489$	0	0
$490$	0	0
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	0	0
$493$	− 9.00000i	− 0.405340i
$494$	−2.00000	−0.0899843
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	0	0
$498$	0	0
$499$	−23.0000	−1.02962	−0.514811	−	0.857304i	$-0.672138\pi$
$499$	−23.0000	−1.02962	−0.514811	+	0.857304i	$0.672138\pi$
$500$	0	0
$501$	0	0
$502$	3.00000i	0.133897i
$503$	− 12.0000i	− 0.535054i	−0.963550	−	0.267527i	$-0.913794\pi$
$503$	− 12.0000i	− 0.535054i	0.963550	−	0.267527i	$-0.0862064\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	− 2.00000i	− 0.0887357i
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	−3.00000	−0.132324
$515$	0	0
$516$	0	0
$517$	0	0
$518$	2.00000i	0.0878750i
$519$	0	0
$520$	0	0
$521$	33.0000	1.44576	0.722878	−	0.690976i	$-0.242819\pi$
$521$	33.0000	1.44576	0.722878	+	0.690976i	$0.242819\pi$
$522$	0	0
$523$	34.0000i	1.48672i	0.668894	+	0.743358i	$0.266768\pi$
$523$	34.0000i	1.48672i	−0.668894	+	0.743358i	$0.733232\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	15.0000	0.654031
$527$	3.00000i	0.130682i
$528$	0	0
$529$	14.0000	0.608696
$530$	0	0
$531$	0	0
$532$	2.00000i	0.0867110i
$533$	3.00000i	0.129944i
$534$	0	0
$535$	0	0
$536$	−8.00000	−0.345547
$537$	0	0
$538$	18.0000i	0.776035i
$539$	0	0
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	− 8.00000i	− 0.343629i
$543$	0	0
$544$	−3.00000	−0.128624
$545$	0	0
$546$	0	0
$547$	− 13.0000i	− 0.555840i	−0.960604	−	0.277920i	$-0.910355\pi$
$547$	− 13.0000i	− 0.555840i	0.960604	−	0.277920i	$-0.0896450\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.00000	−0.255609
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	8.00000	0.339887
$555$	0	0
$556$	−4.00000	−0.169638
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	0	0
$559$	−7.00000	−0.296068
$560$	0	0
$561$	0	0
$562$	24.0000i	1.01238i
$563$	− 45.0000i	− 1.89652i	−0.317489	−	0.948262i	$-0.602840\pi$
$563$	− 45.0000i	− 1.89652i	0.317489	−	0.948262i	$-0.397160\pi$
$564$	0	0
$565$	0	0
$566$	−14.0000	−0.588464
$567$	0	0
$568$	0	0
$569$	−36.0000	−1.50920	−0.754599	−	0.656186i	$-0.772169\pi$
$569$	−36.0000	−1.50920	−0.754599	+	0.656186i	$0.772169\pi$
$570$	0	0
$571$	5.00000	0.209243	0.104622	−	0.994512i	$-0.466637\pi$
$571$	5.00000	0.209243	0.104622	+	0.994512i	$0.466637\pi$
$572$	0	0
$573$	0	0
$574$	3.00000	0.125218
$575$	0	0
$576$	0	0
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	− 8.00000i	− 0.332756i
$579$	0	0
$580$	0	0
$581$	−15.0000	−0.622305
$582$	0	0
$583$	0	0
$584$	−4.00000	−0.165521
$585$	0	0
$586$	12.0000	0.495715
$587$	9.00000i	0.371470i	0.982600	+	0.185735i	$0.0594666\pi$
$587$	9.00000i	0.371470i	−0.982600	+	0.185735i	$0.940533\pi$
$588$	0	0
$589$	2.00000	0.0824086
$590$	0	0
$591$	0	0
$592$	2.00000i	0.0821995i
$593$	− 30.0000i	− 1.23195i	−0.787765	−	0.615976i	$-0.788762\pi$
$593$	− 30.0000i	− 1.23195i	0.787765	−	0.615976i	$-0.211238\pi$
$594$	0	0
$595$	0	0
$596$	−15.0000	−0.614424
$597$	0	0
$598$	3.00000i	0.122679i
$599$	15.0000	0.612883	0.306442	−	0.951889i	$-0.400862\pi$
$599$	15.0000	0.612883	0.306442	+	0.951889i	$0.400862\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	7.00000i	0.285299i
$603$	0	0
$604$	10.0000	0.406894
$605$	0	0
$606$	0	0
$607$	8.00000i	0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$	8.00000i	0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$	2.00000i	0.0811107i
$609$	0	0
$610$	0	0
$611$	6.00000	0.242734
$612$	0	0
$613$	− 26.0000i	− 1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$	− 26.0000i	− 1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	14.0000	0.564994
$615$	0	0
$616$	0	0
$617$	42.0000i	1.69086i	0.534089	+	0.845428i	$0.320655\pi$
$617$	42.0000i	1.69086i	−0.534089	+	0.845428i	$0.679345\pi$
$618$	0	0
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	0	0
$621$	0	0
$622$	− 30.0000i	− 1.20289i
$623$	6.00000i	0.240385i
$624$	0	0
$625$	0	0
$626$	28.0000	1.11911
$627$	0	0
$628$	− 2.00000i	− 0.0798087i
$629$	6.00000	0.239236
$630$	0	0
$631$	26.0000	1.03504	0.517522	−	0.855670i	$-0.326855\pi$
$631$	26.0000	1.03504	0.517522	+	0.855670i	$0.326855\pi$
$632$	− 8.00000i	− 0.318223i
$633$	0	0
$634$	−21.0000	−0.834017
$635$	0	0
$636$	0	0
$637$	− 1.00000i	− 0.0396214i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	0	0
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	3.00000	0.118217
$645$	0	0
$646$	6.00000	0.236067
$647$	− 24.0000i	− 0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$	− 24.0000i	− 0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	− 1.00000i	− 0.0391630i
$653$	− 42.0000i	− 1.64359i	−0.569785	−	0.821794i	$-0.692974\pi$
$653$	− 42.0000i	− 1.64359i	0.569785	−	0.821794i	$-0.307026\pi$
$654$	0	0
$655$	0	0
$656$	3.00000	0.117130
$657$	0	0
$658$	− 6.00000i	− 0.233904i
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	25.0000i	0.971653i
$663$	0	0
$664$	−15.0000	−0.582113
$665$	0	0
$666$	0	0
$667$	9.00000i	0.348481i
$668$	− 12.0000i	− 0.464294i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	25.0000i	0.963679i	0.876259	+	0.481840i	$0.160031\pi$
$673$	25.0000i	0.963679i	−0.876259	+	0.481840i	$0.839969\pi$
$674$	−13.0000	−0.500741
$675$	0	0
$676$	−12.0000	−0.461538
$677$	− 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
$679$	−8.00000	−0.307012
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 6.00000i	− 0.229584i	−0.993390	−	0.114792i	$-0.963380\pi$
$683$	− 6.00000i	− 0.229584i	0.993390	−	0.114792i	$-0.0366201\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	7.00000i	0.266872i
$689$	−9.00000	−0.342873
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	− 6.00000i	− 0.228086i
$693$	0	0
$694$	24.0000	0.911028
$695$	0	0
$696$	0	0
$697$	− 9.00000i	− 0.340899i
$698$	− 31.0000i	− 1.17337i
$699$	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
$703$	− 4.00000i	− 0.150863i
$704$	0	0
$705$	0	0
$706$	−18.0000	−0.677439
$707$	18.0000i	0.676960i
$708$	0	0
$709$	−8.00000	−0.300446	−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000	−0.300446	−0.150223	+	0.988652i	$0.547999\pi$
$710$	0	0
$711$	0	0
$712$	6.00000i	0.224860i
$713$	− 3.00000i	− 0.112351i
$714$	0	0
$715$	0	0
$716$	−6.00000	−0.224231
$717$	0	0
$718$	9.00000i	0.335877i
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	−13.0000	−0.484145
$722$	15.0000i	0.558242i
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	0	0
$727$	35.0000i	1.29808i	0.760755	+	0.649039i	$0.224829\pi$
$727$	35.0000i	1.29808i	−0.760755	+	0.649039i	$0.775171\pi$
$728$	− 1.00000i	− 0.0370625i
$729$	0	0
$730$	0	0
$731$	21.0000	0.776713
$732$	0	0
$733$	− 5.00000i	− 0.184679i	−0.995728	−	0.0923396i	$-0.970565\pi$
$733$	− 5.00000i	− 0.184679i	0.995728	−	0.0923396i	$-0.0294345\pi$
$734$	17.0000	0.627481
$735$	0	0
$736$	3.00000	0.110581
$737$	0	0
$738$	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
$742$	9.00000i	0.330400i
$743$	− 15.0000i	− 0.550297i	−0.961402	−	0.275148i	$-0.911273\pi$
$743$	− 15.0000i	− 0.550297i	0.961402	−	0.275148i	$-0.0887270\pi$
$744$	0	0
$745$	0	0
$746$	−14.0000	−0.512576
$747$	0	0
$748$	0	0
$749$	6.00000	0.219235
$750$	0	0
$751$	−10.0000	−0.364905	−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000	−0.364905	−0.182453	+	0.983215i	$0.558404\pi$
$752$	− 6.00000i	− 0.218797i
$753$	0	0
$754$	3.00000	0.109254
$755$	0	0
$756$	0	0
$757$	− 16.0000i	− 0.581530i	−0.956795	−	0.290765i	$-0.906090\pi$
$757$	− 16.0000i	− 0.581530i	0.956795	−	0.290765i	$-0.0939098\pi$
$758$	5.00000i	0.181608i
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	− 2.00000i	− 0.0724049i
$764$	−9.00000	−0.325609
$765$	0	0
$766$	−12.0000	−0.433578
$767$	3.00000i	0.108324i
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	0	0
$772$	14.0000i	0.503871i
$773$	− 42.0000i	− 1.51064i	−0.655359	−	0.755318i	$-0.727483\pi$
$773$	− 42.0000i	− 1.51064i	0.655359	−	0.755318i	$-0.272517\pi$
$774$	0	0
$775$	0	0
$776$	−8.00000	−0.287183
$777$	0	0
$778$	6.00000i	0.215110i
$779$	−6.00000	−0.214972
$780$	0	0
$781$	0	0
$782$	− 9.00000i	− 0.321839i
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	0	0
$787$	50.0000i	1.78231i	0.453701	+	0.891154i	$0.350103\pi$
$787$	50.0000i	1.78231i	−0.453701	+	0.891154i	$0.649897\pi$
$788$	− 3.00000i	− 0.106871i
$789$	0	0
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	− 1.00000i	− 0.0355110i
$794$	−31.0000	−1.10015
$795$	0	0
$796$	8.00000	0.283552
$797$	36.0000i	1.27519i	0.770374	+	0.637593i	$0.220070\pi$
$797$	36.0000i	1.27519i	−0.770374	+	0.637593i	$0.779930\pi$
$798$	0	0
$799$	−18.0000	−0.636794
$800$	0	0
$801$	0	0
$802$	− 24.0000i	− 0.847469i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−1.00000	−0.0352235
$807$	0	0
$808$	18.0000i	0.633238i
$809$	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
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	− 3.00000i	− 0.105279i
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 14.0000i	− 0.489798i
$818$	− 28.0000i	− 0.978997i
$819$	0	0
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	16.0000i	0.557725i	0.960331	+	0.278862i	$0.0899574\pi$
$823$	16.0000i	0.557725i	−0.960331	+	0.278862i	$0.910043\pi$
$824$	−13.0000	−0.452876
$825$	0	0
$826$	3.00000	0.104383
$827$	30.0000i	1.04320i	0.853189	+	0.521601i	$0.174665\pi$
$827$	30.0000i	1.04320i	−0.853189	+	0.521601i	$0.825335\pi$
$828$	0	0
$829$	−5.00000	−0.173657	−0.0868286	−	0.996223i	$-0.527673\pi$
$829$	−5.00000	−0.173657	−0.0868286	+	0.996223i	$0.527673\pi$
$830$	0	0
$831$	0	0
$832$	− 1.00000i	− 0.0346688i
$833$	3.00000i	0.103944i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	15.0000i	0.518166i
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	16.0000i	0.551396i
$843$	0	0
$844$	7.00000	0.240950
$845$	0	0
$846$	0	0
$847$	− 11.0000i	− 0.377964i
$848$	9.00000i	0.309061i
$849$	0	0
$850$	0	0
$851$	−6.00000	−0.205677
$852$	0	0
$853$	− 17.0000i	− 0.582069i	−0.956713	−	0.291034i	$-0.906001\pi$
$853$	− 17.0000i	− 0.582069i	0.956713	−	0.291034i	$-0.0939994\pi$
$854$	−1.00000	−0.0342193
$855$	0	0
$856$	6.00000	0.205076
$857$	− 42.0000i	− 1.43469i	−0.696717	−	0.717346i	$-0.745357\pi$
$857$	− 42.0000i	− 1.43469i	0.696717	−	0.717346i	$-0.254643\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	0	0
$862$	39.0000i	1.32835i
$863$	− 12.0000i	− 0.408485i	−0.978920	−	0.204242i	$-0.934527\pi$
$863$	− 12.0000i	− 0.408485i	0.978920	−	0.204242i	$-0.0654731\pi$
$864$	0	0
$865$	0	0
$866$	−14.0000	−0.475739
$867$	0	0
$868$	1.00000i	0.0339422i
$869$	0	0
$870$	0	0
$871$	−8.00000	−0.271070
$872$	− 2.00000i	− 0.0677285i
$873$	0	0
$874$	−6.00000	−0.202953
$875$	0	0
$876$	0	0
$877$	− 40.0000i	− 1.35070i	−0.737496	−	0.675352i	$-0.763992\pi$
$877$	− 40.0000i	− 1.35070i	0.737496	−	0.675352i	$-0.236008\pi$
$878$	− 13.0000i	− 0.438729i
$879$	0	0
$880$	0	0
$881$	−33.0000	−1.11180	−0.555899	−	0.831250i	$-0.687626\pi$
$881$	−33.0000	−1.11180	−0.555899	+	0.831250i	$0.687626\pi$
$882$	0	0
$883$	25.0000i	0.841317i	0.907219	+	0.420658i	$0.138201\pi$
$883$	25.0000i	0.841317i	−0.907219	+	0.420658i	$0.861799\pi$
$884$	−3.00000	−0.100901
$885$	0	0
$886$	−18.0000	−0.604722
$887$	18.0000i	0.604381i	0.953248	+	0.302190i	$0.0977178\pi$
$887$	18.0000i	0.604381i	−0.953248	+	0.302190i	$0.902282\pi$
$888$	0	0
$889$	−2.00000	−0.0670778
$890$	0	0
$891$	0	0
$892$	− 13.0000i	− 0.435272i
$893$	12.0000i	0.401565i
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	6.00000i	0.200223i
$899$	−3.00000	−0.100056
$900$	0	0
$901$	27.0000	0.899500
$902$	0	0
$903$	0	0
$904$	−12.0000	−0.399114
$905$	0	0
$906$	0	0
$907$	35.0000i	1.16216i	0.813848	+	0.581078i	$0.197369\pi$
$907$	35.0000i	1.16216i	−0.813848	+	0.581078i	$0.802631\pi$
$908$	21.0000i	0.696909i
$909$	0	0
$910$	0	0
$911$	−57.0000	−1.88849	−0.944247	−	0.329238i	$-0.893208\pi$
$911$	−57.0000	−1.88849	−0.944247	+	0.329238i	$0.893208\pi$
$912$	0	0
$913$	0	0
$914$	−1.00000	−0.0330771
$915$	0	0
$916$	−22.0000	−0.726900
$917$	12.0000i	0.396275i
$918$	0	0
$919$	−50.0000	−1.64935	−0.824674	−	0.565608i	$-0.808641\pi$
$919$	−50.0000	−1.64935	−0.824674	+	0.565608i	$0.808641\pi$
$920$	0	0
$921$	0	0
$922$	30.0000i	0.987997i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	28.0000	0.920137
$927$	0	0
$928$	− 3.00000i	− 0.0984798i
$929$	−27.0000	−0.885841	−0.442921	−	0.896561i	$-0.646058\pi$
$929$	−27.0000	−0.885841	−0.442921	+	0.896561i	$0.646058\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	6.00000i	0.196537i
$933$	0	0
$934$	−3.00000	−0.0981630
$935$	0	0
$936$	0	0
$937$	38.0000i	1.24141i	0.784046	+	0.620703i	$0.213153\pi$
$937$	38.0000i	1.24141i	−0.784046	+	0.620703i	$0.786847\pi$
$938$	8.00000i	0.261209i
$939$	0	0
$940$	0	0
$941$	48.0000	1.56476	0.782378	−	0.622804i	$-0.214007\pi$
$941$	48.0000	1.56476	0.782378	+	0.622804i	$0.214007\pi$
$942$	0	0
$943$	9.00000i	0.293080i
$944$	3.00000	0.0976417
$945$	0	0
$946$	0	0
$947$	− 30.0000i	− 0.974869i	−0.873160	−	0.487435i	$-0.837933\pi$
$947$	− 30.0000i	− 0.974869i	0.873160	−	0.487435i	$-0.162067\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	0	0
$952$	3.00000i	0.0972306i
$953$	36.0000i	1.16615i	0.812417	+	0.583077i	$0.198151\pi$
$953$	36.0000i	1.16615i	−0.812417	+	0.583077i	$0.801849\pi$
$954$	0	0
$955$	0	0
$956$	−24.0000	−0.776215
$957$	0	0
$958$	30.0000i	0.969256i
$959$	0	0
$960$	0	0
$961$	−30.0000	−0.967742
$962$	2.00000i	0.0644826i
$963$	0	0
$964$	22.0000	0.708572
$965$	0	0
$966$	0	0
$967$	− 22.0000i	− 0.707472i	−0.935345	−	0.353736i	$-0.884911\pi$
$967$	− 22.0000i	− 0.707472i	0.935345	−	0.353736i	$-0.115089\pi$
$968$	− 11.0000i	− 0.353553i
$969$	0	0
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	0	0
$973$	4.00000i	0.128234i
$974$	20.0000	0.640841
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	30.0000i	0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$	30.0000i	0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	30.0000i	0.957338i
$983$	− 36.0000i	− 1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$	− 36.0000i	− 1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$	0	0
$985$	0	0
$986$	−9.00000	−0.286618
$987$	0	0
$988$	2.00000i	0.0636285i
$989$	−21.0000	−0.667761
$990$	0	0
$991$	38.0000	1.20711	0.603555	−	0.797321i	$-0.293750\pi$
$991$	38.0000	1.20711	0.603555	+	0.797321i	$0.293750\pi$
$992$	1.00000i	0.0317500i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 46.0000i	− 1.45683i	−0.685134	−	0.728417i	$-0.740256\pi$
$997$	− 46.0000i	− 1.45683i	0.685134	−	0.728417i	$-0.259744\pi$
$998$	23.0000i	0.728052i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.2.g.n.2899.1		2
3.2	odd	2		3150.2.g.k.2899.2		2
5.2	odd	4		3150.2.a.bc.1.1	yes	1
5.3	odd	4		3150.2.a.o.1.1	yes	1
5.4	even	2	inner	3150.2.g.n.2899.2		2
15.2	even	4		3150.2.a.h.1.1	✓	1
15.8	even	4		3150.2.a.bm.1.1	yes	1
15.14	odd	2		3150.2.g.k.2899.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3150.2.a.h.1.1	✓	1	15.2	even	4
3150.2.a.o.1.1	yes	1	5.3	odd	4
3150.2.a.bc.1.1	yes	1	5.2	odd	4
3150.2.a.bm.1.1	yes	1	15.8	even	4
3150.2.g.k.2899.1		2	15.14	odd	2
3150.2.g.k.2899.2		2	3.2	odd	2
3150.2.g.n.2899.1		2	1.1	even	1	trivial
3150.2.g.n.2899.2		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:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +1.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} -2.00000 q^{19} +3.00000i q^{23} +1.00000 q^{26} -1.00000i q^{28} +3.00000 q^{29} -1.00000 q^{31} -1.00000i q^{32} -3.00000 q^{34} +2.00000i q^{37} +2.00000i q^{38} +3.00000 q^{41} +7.00000i q^{43} +3.00000 q^{46} -6.00000i q^{47} -1.00000 q^{49} -1.00000i q^{52} +9.00000i q^{53} -1.00000 q^{56} -3.00000i q^{58} +3.00000 q^{59} -1.00000 q^{61} +1.00000i q^{62} -1.00000 q^{64} +8.00000i q^{67} +3.00000i q^{68} +4.00000i q^{73} +2.00000 q^{74} +2.00000 q^{76} -8.00000 q^{79} -3.00000i q^{82} +15.0000i q^{83} +7.00000 q^{86} +6.00000 q^{89} -1.00000 q^{91} -3.00000i q^{92} -6.00000 q^{94} +8.00000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 2 q^{14} + 2 q^{16} - 4 q^{19} + 2 q^{26} + 6 q^{29} - 2 q^{31} - 6 q^{34} + 6 q^{41} + 6 q^{46} - 2 q^{49} - 2 q^{56} + 6 q^{59} - 2 q^{61} - 2 q^{64} + 4 q^{74} + 4 q^{76} - 16 q^{79} + 14 q^{86} + 12 q^{89} - 2 q^{91} - 12 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^14 + 2 * q^16 - 4 * q^19 + 2 * q^26 + 6 * q^29 - 2 * q^31 - 6 * q^34 + 6 * q^41 + 6 * q^46 - 2 * q^49 - 2 * q^56 + 6 * q^59 - 2 * q^61 - 2 * q^64 + 4 * q^74 + 4 * q^76 - 16 * q^79 + 14 * q^86 + 12 * q^89 - 2 * q^91 - 12 * q^94

Introduction

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Modular forms

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