LMFDB - Embedded newform 675.2.b.i.649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(649,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("675.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	675.b (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:	$5.38990213644$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$-1.30278i$ of defining polynomial
Character	$\chi$	$=$	675.649
Dual form			675.2.b.i.649.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.30278i q^{2} +0.302776 q^{4} -4.60555i q^{7} -3.00000i q^{8} +O(q^{10})$	$q-1.30278i q^{2} +0.302776 q^{4} -4.60555i q^{7} -3.00000i q^{8} -2.60555 q^{11} -0.605551i q^{13} -6.00000 q^{14} -3.30278 q^{16} +5.60555i q^{17} +3.60555 q^{19} +3.39445i q^{22} -3.00000i q^{23} -0.788897 q^{26} -1.39445i q^{28} -8.60555 q^{29} +1.60555 q^{31} -1.69722i q^{32} +7.30278 q^{34} -2.00000i q^{37} -4.69722i q^{38} +2.60555 q^{41} -6.60555i q^{43} -0.788897 q^{44} -3.90833 q^{46} -5.21110i q^{47} -14.2111 q^{49} -0.183346i q^{52} -5.60555i q^{53} -13.8167 q^{56} +11.2111i q^{58} +8.60555 q^{59} +10.2111 q^{61} -2.09167i q^{62} -8.81665 q^{64} +15.2111i q^{67} +1.69722i q^{68} +14.6056 q^{71} +5.39445i q^{73} -2.60555 q^{74} +1.09167 q^{76} +12.0000i q^{77} +4.39445 q^{79} -3.39445i q^{82} +3.00000i q^{83} -8.60555 q^{86} +7.81665i q^{88} -7.81665 q^{89} -2.78890 q^{91} -0.908327i q^{92} -6.78890 q^{94} -8.00000i q^{97} +18.5139i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{4}+O(q^{10}) $ 4 * q - 6 * q^4	$ 4 q - 6 q^{4} + 4 q^{11} - 24 q^{14} - 6 q^{16} - 32 q^{26} - 20 q^{29} - 8 q^{31} + 22 q^{34} - 4 q^{41} - 32 q^{44} + 6 q^{46} - 28 q^{49} - 12 q^{56} + 20 q^{59} + 12 q^{61} + 8 q^{64} + 44 q^{71} + 4 q^{74} + 26 q^{76} + 32 q^{79} - 20 q^{86} + 12 q^{89} - 40 q^{91} - 56 q^{94}+O(q^{100}) $ 4 * q - 6 * q^4 + 4 * q^11 - 24 * q^14 - 6 * q^16 - 32 * q^26 - 20 * q^29 - 8 * q^31 + 22 * q^34 - 4 * q^41 - 32 * q^44 + 6 * q^46 - 28 * q^49 - 12 * q^56 + 20 * q^59 + 12 * q^61 + 8 * q^64 + 44 * q^71 + 4 * q^74 + 26 * q^76 + 32 * q^79 - 20 * q^86 + 12 * q^89 - 40 * q^91 - 56 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.30278i	− 0.921201i	−0.887607	−	0.460601i	$-0.847634\pi$
$2$	− 1.30278i	− 0.921201i	0.887607	−	0.460601i	$-0.152366\pi$
$3$	0	0
$3$	0	0
$4$	0.302776	0.151388
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.60555i	− 1.74073i	−0.492403	−	0.870367i	$-0.663881\pi$
$7$	− 4.60555i	− 1.74073i	0.492403	−	0.870367i	$-0.336119\pi$
$8$	− 3.00000i	− 1.06066i
$9$	0	0
$10$	0	0
$11$	−2.60555	−0.785603	−0.392802	−	0.919623i	$-0.628494\pi$
$11$	−2.60555	−0.785603	−0.392802	+	0.919623i	$0.628494\pi$
$12$	0	0
$13$	− 0.605551i	− 0.167950i	−0.996468	−	0.0839749i	$-0.973238\pi$
$13$	− 0.605551i	− 0.167950i	0.996468	−	0.0839749i	$-0.0267615\pi$
$14$	−6.00000	−1.60357
$15$	0	0
$16$	−3.30278	−0.825694
$17$	5.60555i	1.35955i	0.733423	+	0.679773i	$0.237922\pi$
$17$	5.60555i	1.35955i	−0.733423	+	0.679773i	$0.762078\pi$
$18$	0	0
$19$	3.60555	0.827170	0.413585	−	0.910465i	$-0.364276\pi$
$19$	3.60555	0.827170	0.413585	+	0.910465i	$0.364276\pi$
$20$	0	0
$21$	0	0
$22$	3.39445i	0.723699i
$23$	− 3.00000i	− 0.625543i	−0.949828	−	0.312772i	$-0.898743\pi$
$23$	− 3.00000i	− 0.625543i	0.949828	−	0.312772i	$-0.101257\pi$
$24$	0	0
$25$	0	0
$26$	−0.788897	−0.154716
$27$	0	0
$28$	− 1.39445i	− 0.263526i
$29$	−8.60555	−1.59801	−0.799005	−	0.601324i	$-0.794640\pi$
$29$	−8.60555	−1.59801	−0.799005	+	0.601324i	$0.794640\pi$
$30$	0	0
$31$	1.60555	0.288366	0.144183	−	0.989551i	$-0.453945\pi$
$31$	1.60555	0.288366	0.144183	+	0.989551i	$0.453945\pi$
$32$	− 1.69722i	− 0.300030i
$33$	0	0
$34$	7.30278	1.25242
$35$	0	0
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	− 4.69722i	− 0.761990i
$39$	0	0
$40$	0	0
$41$	2.60555	0.406919	0.203459	−	0.979083i	$-0.434782\pi$
$41$	2.60555	0.406919	0.203459	+	0.979083i	$0.434782\pi$
$42$	0	0
$43$	− 6.60555i	− 1.00734i	−0.863897	−	0.503669i	$-0.831983\pi$
$43$	− 6.60555i	− 1.00734i	0.863897	−	0.503669i	$-0.168017\pi$
$44$	−0.788897	−0.118931
$45$	0	0
$46$	−3.90833	−0.576251
$47$	− 5.21110i	− 0.760117i	−0.924962	−	0.380059i	$-0.875904\pi$
$47$	− 5.21110i	− 0.760117i	0.924962	−	0.380059i	$-0.124096\pi$
$48$	0	0
$49$	−14.2111	−2.03016
$50$	0	0
$51$	0	0
$52$	− 0.183346i	− 0.0254255i
$53$	− 5.60555i	− 0.769982i	−0.922920	−	0.384991i	$-0.874205\pi$
$53$	− 5.60555i	− 0.769982i	0.922920	−	0.384991i	$-0.125795\pi$
$54$	0	0
$55$	0	0
$56$	−13.8167	−1.84633
$57$	0	0
$58$	11.2111i	1.47209i
$59$	8.60555	1.12035	0.560174	−	0.828375i	$-0.310734\pi$
$59$	8.60555	1.12035	0.560174	+	0.828375i	$0.310734\pi$
$60$	0	0
$61$	10.2111	1.30740	0.653699	−	0.756755i	$-0.273216\pi$
$61$	10.2111	1.30740	0.653699	+	0.756755i	$0.273216\pi$
$62$	− 2.09167i	− 0.265643i
$63$	0	0
$64$	−8.81665	−1.10208
$65$	0	0
$66$	0	0
$67$	15.2111i	1.85833i	0.369663	+	0.929166i	$0.379473\pi$
$67$	15.2111i	1.85833i	−0.369663	+	0.929166i	$0.620527\pi$
$68$	1.69722i	0.205819i
$69$	0	0
$70$	0	0
$71$	14.6056	1.73336	0.866680	−	0.498864i	$-0.166249\pi$
$71$	14.6056	1.73336	0.866680	+	0.498864i	$0.166249\pi$
$72$	0	0
$73$	5.39445i	0.631372i	0.948864	+	0.315686i	$0.102235\pi$
$73$	5.39445i	0.631372i	−0.948864	+	0.315686i	$0.897765\pi$
$74$	−2.60555	−0.302889
$75$	0	0
$76$	1.09167	0.125223
$77$	12.0000i	1.36753i
$78$	0	0
$79$	4.39445	0.494414	0.247207	−	0.968963i	$-0.420487\pi$
$79$	4.39445	0.494414	0.247207	+	0.968963i	$0.420487\pi$
$80$	0	0
$81$	0	0
$82$	− 3.39445i	− 0.374854i
$83$	3.00000i	0.329293i	0.986353	+	0.164646i	$0.0526483\pi$
$83$	3.00000i	0.329293i	−0.986353	+	0.164646i	$0.947352\pi$
$84$	0	0
$85$	0	0
$86$	−8.60555	−0.927960
$87$	0	0
$88$	7.81665i	0.833258i
$89$	−7.81665	−0.828564	−0.414282	−	0.910149i	$-0.635967\pi$
$89$	−7.81665	−0.828564	−0.414282	+	0.910149i	$0.635967\pi$
$90$	0	0
$91$	−2.78890	−0.292356
$92$	− 0.908327i	− 0.0946996i
$93$	0	0
$94$	−6.78890	−0.700221
$95$	0	0
$96$	0	0
$97$	− 8.00000i	− 0.812277i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	− 8.00000i	− 0.812277i	0.913812	−	0.406138i	$-0.133125\pi$
$98$	18.5139i	1.87018i
$99$	0	0
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	−1.81665	−0.178138
$105$	0	0
$106$	−7.30278	−0.709308
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	0	0
$111$	0	0
$112$	15.2111i	1.43731i
$113$	− 0.788897i	− 0.0742132i	−0.999311	−	0.0371066i	$-0.988186\pi$
$113$	− 0.788897i	− 0.0742132i	0.999311	−	0.0371066i	$-0.0118141\pi$
$114$	0	0
$115$	0	0
$116$	−2.60555	−0.241919
$117$	0	0
$118$	− 11.2111i	− 1.03207i
$119$	25.8167	2.36661
$120$	0	0
$121$	−4.21110	−0.382828
$122$	− 13.3028i	− 1.20438i
$123$	0	0
$124$	0.486122	0.0436550
$125$	0	0
$126$	0	0
$127$	4.78890i	0.424946i	0.977167	+	0.212473i	$0.0681517\pi$
$127$	4.78890i	0.424946i	−0.977167	+	0.212473i	$0.931848\pi$
$128$	8.09167i	0.715210i
$129$	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
$133$	− 16.6056i	− 1.43988i
$134$	19.8167	1.71190
$135$	0	0
$136$	16.8167	1.44202
$137$	4.81665i	0.411515i	0.978603	+	0.205757i	$0.0659657\pi$
$137$	4.81665i	0.411515i	−0.978603	+	0.205757i	$0.934034\pi$
$138$	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
$142$	− 19.0278i	− 1.59677i
$143$	1.57779i	0.131942i
$144$	0	0
$145$	0	0
$146$	7.02776	0.581621
$147$	0	0
$148$	− 0.605551i	− 0.0497760i
$149$	13.0278	1.06728	0.533638	−	0.845713i	$-0.320825\pi$
$149$	13.0278	1.06728	0.533638	+	0.845713i	$0.320825\pi$
$150$	0	0
$151$	−14.4222	−1.17366	−0.586831	−	0.809709i	$-0.699625\pi$
$151$	−14.4222	−1.17366	−0.586831	+	0.809709i	$0.699625\pi$
$152$	− 10.8167i	− 0.877346i
$153$	0	0
$154$	15.6333	1.25977
$155$	0	0
$156$	0	0
$157$	− 3.81665i	− 0.304602i	−0.988334	−	0.152301i	$-0.951332\pi$
$157$	− 3.81665i	− 0.304602i	0.988334	−	0.152301i	$-0.0486683\pi$
$158$	− 5.72498i	− 0.455455i
$159$	0	0
$160$	0	0
$161$	−13.8167	−1.08890
$162$	0	0
$163$	2.00000i	0.156652i	0.996928	+	0.0783260i	$0.0249575\pi$
$163$	2.00000i	0.156652i	−0.996928	+	0.0783260i	$0.975042\pi$
$164$	0.788897	0.0616025
$165$	0	0
$166$	3.90833	0.303345
$167$	3.00000i	0.232147i	0.993241	+	0.116073i	$0.0370308\pi$
$167$	3.00000i	0.232147i	−0.993241	+	0.116073i	$0.962969\pi$
$168$	0	0
$169$	12.6333	0.971793
$170$	0	0
$171$	0	0
$172$	− 2.00000i	− 0.152499i
$173$	10.8167i	0.822375i	0.911551	+	0.411187i	$0.134886\pi$
$173$	10.8167i	0.822375i	−0.911551	+	0.411187i	$0.865114\pi$
$174$	0	0
$175$	0	0
$176$	8.60555	0.648668
$177$	0	0
$178$	10.1833i	0.763274i
$179$	6.78890	0.507426	0.253713	−	0.967280i	$-0.418348\pi$
$179$	6.78890	0.507426	0.253713	+	0.967280i	$0.418348\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	3.63331i	0.269319i
$183$	0	0
$184$	−9.00000	−0.663489
$185$	0	0
$186$	0	0
$187$	− 14.6056i	− 1.06806i
$188$	− 1.57779i	− 0.115073i
$189$	0	0
$190$	0	0
$191$	16.4222	1.18827	0.594135	−	0.804366i	$-0.297495\pi$
$191$	16.4222	1.18827	0.594135	+	0.804366i	$0.297495\pi$
$192$	0	0
$193$	21.8167i	1.57040i	0.619244	+	0.785199i	$0.287439\pi$
$193$	21.8167i	1.57040i	−0.619244	+	0.785199i	$0.712561\pi$
$194$	−10.4222	−0.748271
$195$	0	0
$196$	−4.30278	−0.307341
$197$	1.18335i	0.0843099i	0.999111	+	0.0421550i	$0.0134223\pi$
$197$	1.18335i	0.0843099i	−0.999111	+	0.0421550i	$0.986578\pi$
$198$	0	0
$199$	−13.2111	−0.936510	−0.468255	−	0.883593i	$-0.655117\pi$
$199$	−13.2111	−0.936510	−0.468255	+	0.883593i	$0.655117\pi$
$200$	0	0
$201$	0	0
$202$	− 15.6333i	− 1.09996i
$203$	39.6333i	2.78171i
$204$	0	0
$205$	0	0
$206$	−5.21110	−0.363075
$207$	0	0
$208$	2.00000i	0.138675i
$209$	−9.39445	−0.649828
$210$	0	0
$211$	12.8167	0.882335	0.441167	−	0.897425i	$-0.354564\pi$
$211$	12.8167	0.882335	0.441167	+	0.897425i	$0.354564\pi$
$212$	− 1.69722i	− 0.116566i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 7.39445i	− 0.501968i
$218$	− 9.11943i	− 0.617646i
$219$	0	0
$220$	0	0
$221$	3.39445	0.228335
$222$	0	0
$223$	− 10.0000i	− 0.669650i	−0.942280	−	0.334825i	$-0.891323\pi$
$223$	− 10.0000i	− 0.669650i	0.942280	−	0.334825i	$-0.108677\pi$
$224$	−7.81665	−0.522272
$225$	0	0
$226$	−1.02776	−0.0683653
$227$	26.2111i	1.73969i	0.493323	+	0.869846i	$0.335782\pi$
$227$	26.2111i	1.73969i	−0.493323	+	0.869846i	$0.664218\pi$
$228$	0	0
$229$	6.21110	0.410441	0.205221	−	0.978716i	$-0.434209\pi$
$229$	6.21110	0.410441	0.205221	+	0.978716i	$0.434209\pi$
$230$	0	0
$231$	0	0
$232$	25.8167i	1.69495i
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	0	0
$235$	0	0
$236$	2.60555	0.169607
$237$	0	0
$238$	− 33.6333i	− 2.18012i
$239$	0.788897	0.0510295	0.0255148	−	0.999674i	$-0.491878\pi$
$239$	0.788897	0.0510295	0.0255148	+	0.999674i	$0.491878\pi$
$240$	0	0
$241$	28.2111	1.81724	0.908618	−	0.417627i	$-0.137138\pi$
$241$	28.2111	1.81724	0.908618	+	0.417627i	$0.137138\pi$
$242$	5.48612i	0.352661i
$243$	0	0
$244$	3.09167	0.197924
$245$	0	0
$246$	0	0
$247$	− 2.18335i	− 0.138923i
$248$	− 4.81665i	− 0.305858i
$249$	0	0
$250$	0	0
$251$	−15.6333	−0.986766	−0.493383	−	0.869812i	$-0.664240\pi$
$251$	−15.6333	−0.986766	−0.493383	+	0.869812i	$0.664240\pi$
$252$	0	0
$253$	7.81665i	0.491429i
$254$	6.23886	0.391461
$255$	0	0
$256$	−7.09167	−0.443230
$257$	− 22.8167i	− 1.42326i	−0.702553	−	0.711632i	$-0.747956\pi$
$257$	− 22.8167i	− 1.42326i	0.702553	−	0.711632i	$-0.252044\pi$
$258$	0	0
$259$	−9.21110	−0.572350
$260$	0	0
$261$	0	0
$262$	− 7.81665i	− 0.482914i
$263$	− 17.2111i	− 1.06128i	−0.847597	−	0.530641i	$-0.821951\pi$
$263$	− 17.2111i	− 1.06128i	0.847597	−	0.530641i	$-0.178049\pi$
$264$	0	0
$265$	0	0
$266$	−21.6333	−1.32642
$267$	0	0
$268$	4.60555i	0.281329i
$269$	−11.2111	−0.683553	−0.341776	−	0.939781i	$-0.611029\pi$
$269$	−11.2111	−0.683553	−0.341776	+	0.939781i	$0.611029\pi$
$270$	0	0
$271$	−19.2389	−1.16868	−0.584339	−	0.811510i	$-0.698646\pi$
$271$	−19.2389	−1.16868	−0.584339	+	0.811510i	$0.698646\pi$
$272$	− 18.5139i	− 1.12257i
$273$	0	0
$274$	6.27502	0.379088
$275$	0	0
$276$	0	0
$277$	29.0278i	1.74411i	0.489409	+	0.872054i	$0.337213\pi$
$277$	29.0278i	1.74411i	−0.489409	+	0.872054i	$0.662787\pi$
$278$	− 5.21110i	− 0.312541i
$279$	0	0
$280$	0	0
$281$	1.81665	0.108372	0.0541862	−	0.998531i	$-0.482744\pi$
$281$	1.81665	0.108372	0.0541862	+	0.998531i	$0.482744\pi$
$282$	0	0
$283$	10.6056i	0.630435i	0.949020	+	0.315217i	$0.102077\pi$
$283$	10.6056i	0.630435i	−0.949020	+	0.315217i	$0.897923\pi$
$284$	4.42221	0.262410
$285$	0	0
$286$	2.05551	0.121545
$287$	− 12.0000i	− 0.708338i
$288$	0	0
$289$	−14.4222	−0.848365
$290$	0	0
$291$	0	0
$292$	1.63331i	0.0955821i
$293$	− 28.8167i	− 1.68349i	−0.539878	−	0.841743i	$-0.681530\pi$
$293$	− 28.8167i	− 1.68349i	0.539878	−	0.841743i	$-0.318470\pi$
$294$	0	0
$295$	0	0
$296$	−6.00000	−0.348743
$297$	0	0
$298$	− 16.9722i	− 0.983176i
$299$	−1.81665	−0.105060
$300$	0	0
$301$	−30.4222	−1.75351
$302$	18.7889i	1.08118i
$303$	0	0
$304$	−11.9083	−0.682989
$305$	0	0
$306$	0	0
$307$	20.4222i	1.16556i	0.812631	+	0.582778i	$0.198034\pi$
$307$	20.4222i	1.16556i	−0.812631	+	0.582778i	$0.801966\pi$
$308$	3.63331i	0.207027i
$309$	0	0
$310$	0	0
$311$	13.8167	0.783471	0.391735	−	0.920078i	$-0.371875\pi$
$311$	13.8167	0.783471	0.391735	+	0.920078i	$0.371875\pi$
$312$	0	0
$313$	23.6333i	1.33583i	0.744236	+	0.667917i	$0.232814\pi$
$313$	23.6333i	1.33583i	−0.744236	+	0.667917i	$0.767186\pi$
$314$	−4.97224	−0.280600
$315$	0	0
$316$	1.33053	0.0748483
$317$	0.394449i	0.0221544i	0.999939	+	0.0110772i	$0.00352606\pi$
$317$	0.394449i	0.0221544i	−0.999939	+	0.0110772i	$0.996474\pi$
$318$	0	0
$319$	22.4222	1.25540
$320$	0	0
$321$	0	0
$322$	18.0000i	1.00310i
$323$	20.2111i	1.12458i
$324$	0	0
$325$	0	0
$326$	2.60555	0.144308
$327$	0	0
$328$	− 7.81665i	− 0.431603i
$329$	−24.0000	−1.32316
$330$	0	0
$331$	14.7889	0.812871	0.406436	−	0.913679i	$-0.366772\pi$
$331$	14.7889	0.812871	0.406436	+	0.913679i	$0.366772\pi$
$332$	0.908327i	0.0498509i
$333$	0	0
$334$	3.90833	0.213854
$335$	0	0
$336$	0	0
$337$	0.605551i	0.0329865i	0.999864	+	0.0164932i	$0.00525020\pi$
$337$	0.605551i	0.0329865i	−0.999864	+	0.0164932i	$0.994750\pi$
$338$	− 16.4584i	− 0.895217i
$339$	0	0
$340$	0	0
$341$	−4.18335	−0.226541
$342$	0	0
$343$	33.2111i	1.79323i
$344$	−19.8167	−1.06844
$345$	0	0
$346$	14.0917	0.757573
$347$	− 1.57779i	− 0.0847005i	−0.999103	−	0.0423502i	$-0.986515\pi$
$347$	− 1.57779i	− 0.0847005i	0.999103	−	0.0423502i	$-0.0134845\pi$
$348$	0	0
$349$	−25.8444	−1.38342	−0.691710	−	0.722176i	$-0.743142\pi$
$349$	−25.8444	−1.38342	−0.691710	+	0.722176i	$0.743142\pi$
$350$	0	0
$351$	0	0
$352$	4.42221i	0.235704i
$353$	− 21.6333i	− 1.15142i	−0.817652	−	0.575712i	$-0.804725\pi$
$353$	− 21.6333i	− 1.15142i	0.817652	−	0.575712i	$-0.195275\pi$
$354$	0	0
$355$	0	0
$356$	−2.36669	−0.125434
$357$	0	0
$358$	− 8.84441i	− 0.467442i
$359$	−33.6333	−1.77510	−0.887549	−	0.460713i	$-0.847594\pi$
$359$	−33.6333	−1.77510	−0.887549	+	0.460713i	$0.847594\pi$
$360$	0	0
$361$	−6.00000	−0.315789
$362$	9.11943i	0.479307i
$363$	0	0
$364$	−0.844410	−0.0442591
$365$	0	0
$366$	0	0
$367$	− 4.60555i	− 0.240408i	−0.992749	−	0.120204i	$-0.961645\pi$
$367$	− 4.60555i	− 0.240408i	0.992749	−	0.120204i	$-0.0383548\pi$
$368$	9.90833i	0.516507i
$369$	0	0
$370$	0	0
$371$	−25.8167	−1.34033
$372$	0	0
$373$	− 10.7889i	− 0.558628i	−0.960200	−	0.279314i	$-0.909893\pi$
$373$	− 10.7889i	− 0.558628i	0.960200	−	0.279314i	$-0.0901070\pi$
$374$	−19.0278	−0.983902
$375$	0	0
$376$	−15.6333	−0.806226
$377$	5.21110i	0.268385i
$378$	0	0
$379$	−14.3944	−0.739393	−0.369697	−	0.929153i	$-0.620538\pi$
$379$	−14.3944	−0.739393	−0.369697	+	0.929153i	$0.620538\pi$
$380$	0	0
$381$	0	0
$382$	− 21.3944i	− 1.09464i
$383$	− 18.6333i	− 0.952118i	−0.879413	−	0.476059i	$-0.842065\pi$
$383$	− 18.6333i	− 0.952118i	0.879413	−	0.476059i	$-0.157935\pi$
$384$	0	0
$385$	0	0
$386$	28.4222	1.44665
$387$	0	0
$388$	− 2.42221i	− 0.122969i
$389$	−4.18335	−0.212104	−0.106052	−	0.994361i	$-0.533821\pi$
$389$	−4.18335	−0.212104	−0.106052	+	0.994361i	$0.533821\pi$
$390$	0	0
$391$	16.8167	0.850455
$392$	42.6333i	2.15331i
$393$	0	0
$394$	1.54163	0.0776664
$395$	0	0
$396$	0	0
$397$	12.6056i	0.632654i	0.948650	+	0.316327i	$0.102450\pi$
$397$	12.6056i	0.632654i	−0.948650	+	0.316327i	$0.897550\pi$
$398$	17.2111i	0.862715i
$399$	0	0
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	− 0.972244i	− 0.0484309i
$404$	3.63331	0.180764
$405$	0	0
$406$	51.6333	2.56252
$407$	5.21110i	0.258305i
$408$	0	0
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	0	0
$411$	0	0
$412$	− 1.21110i	− 0.0596667i
$413$	− 39.6333i	− 1.95023i
$414$	0	0
$415$	0	0
$416$	−1.02776	−0.0503899
$417$	0	0
$418$	12.2389i	0.598622i
$419$	13.0278	0.636448	0.318224	−	0.948016i	$-0.396914\pi$
$419$	13.0278	0.636448	0.318224	+	0.948016i	$0.396914\pi$
$420$	0	0
$421$	−23.4222	−1.14153	−0.570764	−	0.821114i	$-0.693353\pi$
$421$	−23.4222	−1.14153	−0.570764	+	0.821114i	$0.693353\pi$
$422$	− 16.6972i	− 0.812808i
$423$	0	0
$424$	−16.8167	−0.816689
$425$	0	0
$426$	0	0
$427$	− 47.0278i	− 2.27583i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	25.8167	1.24354	0.621772	−	0.783198i	$-0.286413\pi$
$431$	25.8167	1.24354	0.621772	+	0.783198i	$0.286413\pi$
$432$	0	0
$433$	− 28.2389i	− 1.35707i	−0.734567	−	0.678536i	$-0.762615\pi$
$433$	− 28.2389i	− 1.35707i	0.734567	−	0.678536i	$-0.237385\pi$
$434$	−9.63331	−0.462414
$435$	0	0
$436$	2.11943	0.101502
$437$	− 10.8167i	− 0.517431i
$438$	0	0
$439$	−20.3944	−0.973374	−0.486687	−	0.873576i	$-0.661795\pi$
$439$	−20.3944	−0.973374	−0.486687	+	0.873576i	$0.661795\pi$
$440$	0	0
$441$	0	0
$442$	− 4.42221i	− 0.210343i
$443$	18.6333i	0.885295i	0.896696	+	0.442648i	$0.145961\pi$
$443$	18.6333i	0.885295i	−0.896696	+	0.442648i	$0.854039\pi$
$444$	0	0
$445$	0	0
$446$	−13.0278	−0.616882
$447$	0	0
$448$	40.6056i	1.91843i
$449$	12.2389	0.577587	0.288794	−	0.957391i	$-0.406746\pi$
$449$	12.2389	0.577587	0.288794	+	0.957391i	$0.406746\pi$
$450$	0	0
$451$	−6.78890	−0.319677
$452$	− 0.238859i	− 0.0112350i
$453$	0	0
$454$	34.1472	1.60261
$455$	0	0
$456$	0	0
$457$	− 1.21110i	− 0.0566530i	−0.999599	−	0.0283265i	$-0.990982\pi$
$457$	− 1.21110i	− 0.0566530i	0.999599	−	0.0283265i	$-0.00901781\pi$
$458$	− 8.09167i	− 0.378099i
$459$	0	0
$460$	0	0
$461$	−21.6333	−1.00756	−0.503782	−	0.863831i	$-0.668058\pi$
$461$	−21.6333	−1.00756	−0.503782	+	0.863831i	$0.668058\pi$
$462$	0	0
$463$	− 15.2111i	− 0.706920i	−0.935450	−	0.353460i	$-0.885005\pi$
$463$	− 15.2111i	− 0.706920i	0.935450	−	0.353460i	$-0.114995\pi$
$464$	28.4222	1.31947
$465$	0	0
$466$	−23.4500	−1.08630
$467$	2.21110i	0.102318i	0.998691	+	0.0511588i	$0.0162915\pi$
$467$	2.21110i	0.102318i	−0.998691	+	0.0511588i	$0.983709\pi$
$468$	0	0
$469$	70.0555	3.23486
$470$	0	0
$471$	0	0
$472$	− 25.8167i	− 1.18831i
$473$	17.2111i	0.791367i
$474$	0	0
$475$	0	0
$476$	7.81665	0.358276
$477$	0	0
$478$	− 1.02776i	− 0.0470085i
$479$	16.1833	0.739436	0.369718	−	0.929144i	$-0.379454\pi$
$479$	16.1833	0.739436	0.369718	+	0.929144i	$0.379454\pi$
$480$	0	0
$481$	−1.21110	−0.0552215
$482$	− 36.7527i	− 1.67404i
$483$	0	0
$484$	−1.27502	−0.0579554
$485$	0	0
$486$	0	0
$487$	8.18335i	0.370823i	0.982661	+	0.185411i	$0.0593618\pi$
$487$	8.18335i	0.370823i	−0.982661	+	0.185411i	$0.940638\pi$
$488$	− 30.6333i	− 1.38670i
$489$	0	0
$490$	0	0
$491$	12.7889	0.577155	0.288577	−	0.957457i	$-0.406818\pi$
$491$	12.7889	0.577155	0.288577	+	0.957457i	$0.406818\pi$
$492$	0	0
$493$	− 48.2389i	− 2.17257i
$494$	−2.84441	−0.127976
$495$	0	0
$496$	−5.30278	−0.238102
$497$	− 67.2666i	− 3.01732i
$498$	0	0
$499$	27.6056	1.23579	0.617897	−	0.786259i	$-0.287985\pi$
$499$	27.6056	1.23579	0.617897	+	0.786259i	$0.287985\pi$
$500$	0	0
$501$	0	0
$502$	20.3667i	0.909010i
$503$	31.4222i	1.40105i	0.713629	+	0.700523i	$0.247050\pi$
$503$	31.4222i	1.40105i	−0.713629	+	0.700523i	$0.752950\pi$
$504$	0	0
$505$	0	0
$506$	10.1833	0.452705
$507$	0	0
$508$	1.44996i	0.0643316i
$509$	−32.8444	−1.45580	−0.727901	−	0.685682i	$-0.759504\pi$
$509$	−32.8444	−1.45580	−0.727901	+	0.685682i	$0.759504\pi$
$510$	0	0
$511$	24.8444	1.09905
$512$	25.4222i	1.12351i
$513$	0	0
$514$	−29.7250	−1.31111
$515$	0	0
$516$	0	0
$517$	13.5778i	0.597151i
$518$	12.0000i	0.527250i
$519$	0	0
$520$	0	0
$521$	−21.3944	−0.937308	−0.468654	−	0.883382i	$-0.655261\pi$
$521$	−21.3944	−0.937308	−0.468654	+	0.883382i	$0.655261\pi$
$522$	0	0
$523$	23.3944i	1.02297i	0.859293	+	0.511484i	$0.170904\pi$
$523$	23.3944i	1.02297i	−0.859293	+	0.511484i	$0.829096\pi$
$524$	1.81665	0.0793609
$525$	0	0
$526$	−22.4222	−0.977655
$527$	9.00000i	0.392046i
$528$	0	0
$529$	14.0000	0.608696
$530$	0	0
$531$	0	0
$532$	− 5.02776i	− 0.217981i
$533$	− 1.57779i	− 0.0683419i
$534$	0	0
$535$	0	0
$536$	45.6333	1.97106
$537$	0	0
$538$	14.6056i	0.629690i
$539$	37.0278	1.59490
$540$	0	0
$541$	−11.5778	−0.497768	−0.248884	−	0.968533i	$-0.580064\pi$
$541$	−11.5778	−0.497768	−0.248884	+	0.968533i	$0.580064\pi$
$542$	25.0639i	1.07659i
$543$	0	0
$544$	9.51388	0.407904
$545$	0	0
$546$	0	0
$547$	6.60555i	0.282433i	0.989979	+	0.141216i	$0.0451014\pi$
$547$	6.60555i	0.282433i	−0.989979	+	0.141216i	$0.954899\pi$
$548$	1.45837i	0.0622983i
$549$	0	0
$550$	0	0
$551$	−31.0278	−1.32183
$552$	0	0
$553$	− 20.2389i	− 0.860644i
$554$	37.8167	1.60668
$555$	0	0
$556$	1.21110	0.0513622
$557$	33.6333i	1.42509i	0.701627	+	0.712544i	$0.252457\pi$
$557$	33.6333i	1.42509i	−0.701627	+	0.712544i	$0.747543\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	− 2.36669i	− 0.0998329i
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	0	0
$565$	0	0
$566$	13.8167	0.580757
$567$	0	0
$568$	− 43.8167i	− 1.83851i
$569$	−37.8167	−1.58536	−0.792678	−	0.609640i	$-0.791314\pi$
$569$	−37.8167	−1.58536	−0.792678	+	0.609640i	$0.791314\pi$
$570$	0	0
$571$	−36.4500	−1.52538	−0.762692	−	0.646762i	$-0.776123\pi$
$571$	−36.4500	−1.52538	−0.762692	+	0.646762i	$0.776123\pi$
$572$	0.477718i	0.0199744i
$573$	0	0
$574$	−15.6333	−0.652522
$575$	0	0
$576$	0	0
$577$	− 27.8167i	− 1.15802i	−0.815320	−	0.579011i	$-0.803439\pi$
$577$	− 27.8167i	− 1.15802i	0.815320	−	0.579011i	$-0.196561\pi$
$578$	18.7889i	0.781515i
$579$	0	0
$580$	0	0
$581$	13.8167	0.573211
$582$	0	0
$583$	14.6056i	0.604900i
$584$	16.1833	0.669672
$585$	0	0
$586$	−37.5416	−1.55083
$587$	− 21.0000i	− 0.866763i	−0.901211	−	0.433381i	$-0.857320\pi$
$587$	− 21.0000i	− 0.866763i	0.901211	−	0.433381i	$-0.142680\pi$
$588$	0	0
$589$	5.78890	0.238527
$590$	0	0
$591$	0	0
$592$	6.60555i	0.271486i
$593$	21.2389i	0.872175i	0.899904	+	0.436088i	$0.143636\pi$
$593$	21.2389i	0.872175i	−0.899904	+	0.436088i	$0.856364\pi$
$594$	0	0
$595$	0	0
$596$	3.94449	0.161572
$597$	0	0
$598$	2.36669i	0.0967812i
$599$	15.3944	0.629000	0.314500	−	0.949257i	$-0.398163\pi$
$599$	15.3944	0.629000	0.314500	+	0.949257i	$0.398163\pi$
$600$	0	0
$601$	32.6333	1.33114	0.665570	−	0.746335i	$-0.268188\pi$
$601$	32.6333	1.33114	0.665570	+	0.746335i	$0.268188\pi$
$602$	39.6333i	1.61533i
$603$	0	0
$604$	−4.36669	−0.177678
$605$	0	0
$606$	0	0
$607$	− 17.3944i	− 0.706019i	−0.935620	−	0.353009i	$-0.885158\pi$
$607$	− 17.3944i	− 0.706019i	0.935620	−	0.353009i	$-0.114842\pi$
$608$	− 6.11943i	− 0.248176i
$609$	0	0
$610$	0	0
$611$	−3.15559	−0.127661
$612$	0	0
$613$	28.8444i	1.16501i	0.812825	+	0.582507i	$0.197928\pi$
$613$	28.8444i	1.16501i	−0.812825	+	0.582507i	$0.802072\pi$
$614$	26.6056	1.07371
$615$	0	0
$616$	36.0000	1.45048
$617$	26.4500i	1.06484i	0.846482	+	0.532418i	$0.178716\pi$
$617$	26.4500i	1.06484i	−0.846482	+	0.532418i	$0.821284\pi$
$618$	0	0
$619$	7.63331	0.306809	0.153404	−	0.988164i	$-0.450976\pi$
$619$	7.63331	0.306809	0.153404	+	0.988164i	$0.450976\pi$
$620$	0	0
$621$	0	0
$622$	− 18.0000i	− 0.721734i
$623$	36.0000i	1.44231i
$624$	0	0
$625$	0	0
$626$	30.7889	1.23057
$627$	0	0
$628$	− 1.15559i	− 0.0461131i
$629$	11.2111	0.447016
$630$	0	0
$631$	30.0278	1.19539	0.597693	−	0.801725i	$-0.296084\pi$
$631$	30.0278	1.19539	0.597693	+	0.801725i	$0.296084\pi$
$632$	− 13.1833i	− 0.524405i
$633$	0	0
$634$	0.513878	0.0204087
$635$	0	0
$636$	0	0
$637$	8.60555i	0.340964i
$638$	− 29.2111i	− 1.15648i
$639$	0	0
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	− 23.0278i	− 0.908126i	−0.890970	−	0.454063i	$-0.849974\pi$
$643$	− 23.0278i	− 0.908126i	0.890970	−	0.454063i	$-0.150026\pi$
$644$	−4.18335	−0.164847
$645$	0	0
$646$	26.3305	1.03596
$647$	− 38.2111i	− 1.50223i	−0.660169	−	0.751117i	$-0.729516\pi$
$647$	− 38.2111i	− 1.50223i	0.660169	−	0.751117i	$-0.270484\pi$
$648$	0	0
$649$	−22.4222	−0.880149
$650$	0	0
$651$	0	0
$652$	0.605551i	0.0237152i
$653$	27.2389i	1.06594i	0.846134	+	0.532969i	$0.178924\pi$
$653$	27.2389i	1.06594i	−0.846134	+	0.532969i	$0.821076\pi$
$654$	0	0
$655$	0	0
$656$	−8.60555	−0.335990
$657$	0	0
$658$	31.2666i	1.21890i
$659$	20.6056	0.802678	0.401339	−	0.915930i	$-0.368545\pi$
$659$	20.6056	0.802678	0.401339	+	0.915930i	$0.368545\pi$
$660$	0	0
$661$	22.8444	0.888545	0.444272	−	0.895892i	$-0.353462\pi$
$661$	22.8444	0.888545	0.444272	+	0.895892i	$0.353462\pi$
$662$	− 19.2666i	− 0.748818i
$663$	0	0
$664$	9.00000	0.349268
$665$	0	0
$666$	0	0
$667$	25.8167i	0.999625i
$668$	0.908327i	0.0351442i
$669$	0	0
$670$	0	0
$671$	−26.6056	−1.02710
$672$	0	0
$673$	− 11.0278i	− 0.425089i	−0.977151	−	0.212544i	$-0.931825\pi$
$673$	− 11.0278i	− 0.425089i	0.977151	−	0.212544i	$-0.0681750\pi$
$674$	0.788897	0.0303872
$675$	0	0
$676$	3.82506	0.147118
$677$	21.6333i	0.831436i	0.909494	+	0.415718i	$0.136470\pi$
$677$	21.6333i	0.831436i	−0.909494	+	0.415718i	$0.863530\pi$
$678$	0	0
$679$	−36.8444	−1.41396
$680$	0	0
$681$	0	0
$682$	5.44996i	0.208690i
$683$	− 9.00000i	− 0.344375i	−0.985064	−	0.172188i	$-0.944916\pi$
$683$	− 9.00000i	− 0.344375i	0.985064	−	0.172188i	$-0.0550836\pi$
$684$	0	0
$685$	0	0
$686$	43.2666	1.65193
$687$	0	0
$688$	21.8167i	0.831752i
$689$	−3.39445	−0.129318
$690$	0	0
$691$	2.39445	0.0910891	0.0455446	−	0.998962i	$-0.485498\pi$
$691$	2.39445	0.0910891	0.0455446	+	0.998962i	$0.485498\pi$
$692$	3.27502i	0.124498i
$693$	0	0
$694$	−2.05551	−0.0780262
$695$	0	0
$696$	0	0
$697$	14.6056i	0.553225i
$698$	33.6695i	1.27441i
$699$	0	0
$700$	0	0
$701$	40.4222	1.52673	0.763363	−	0.645970i	$-0.223547\pi$
$701$	40.4222	1.52673	0.763363	+	0.645970i	$0.223547\pi$
$702$	0	0
$703$	− 7.21110i	− 0.271972i
$704$	22.9722	0.865799
$705$	0	0
$706$	−28.1833	−1.06069
$707$	− 55.2666i	− 2.07851i
$708$	0	0
$709$	−34.8444	−1.30861	−0.654305	−	0.756231i	$-0.727039\pi$
$709$	−34.8444	−1.30861	−0.654305	+	0.756231i	$0.727039\pi$
$710$	0	0
$711$	0	0
$712$	23.4500i	0.878824i
$713$	− 4.81665i	− 0.180385i
$714$	0	0
$715$	0	0
$716$	2.05551	0.0768181
$717$	0	0
$718$	43.8167i	1.63522i
$719$	49.2666	1.83733	0.918667	−	0.395032i	$-0.129267\pi$
$719$	49.2666	1.83733	0.918667	+	0.395032i	$0.129267\pi$
$720$	0	0
$721$	−18.4222	−0.686079
$722$	7.81665i	0.290906i
$723$	0	0
$724$	−2.11943	−0.0787680
$725$	0	0
$726$	0	0
$727$	7.63331i	0.283104i	0.989931	+	0.141552i	$0.0452092\pi$
$727$	7.63331i	0.283104i	−0.989931	+	0.141552i	$0.954791\pi$
$728$	8.36669i	0.310090i
$729$	0	0
$730$	0	0
$731$	37.0278	1.36952
$732$	0	0
$733$	32.0000i	1.18195i	0.806691	+	0.590973i	$0.201256\pi$
$733$	32.0000i	1.18195i	−0.806691	+	0.590973i	$0.798744\pi$
$734$	−6.00000	−0.221464
$735$	0	0
$736$	−5.09167	−0.187682
$737$	− 39.6333i	− 1.45991i
$738$	0	0
$739$	−30.0278	−1.10459	−0.552294	−	0.833649i	$-0.686248\pi$
$739$	−30.0278	−1.10459	−0.552294	+	0.833649i	$0.686248\pi$
$740$	0	0
$741$	0	0
$742$	33.6333i	1.23472i
$743$	− 34.4222i	− 1.26283i	−0.775446	−	0.631414i	$-0.782475\pi$
$743$	− 34.4222i	− 1.26283i	0.775446	−	0.631414i	$-0.217525\pi$
$744$	0	0
$745$	0	0
$746$	−14.0555	−0.514609
$747$	0	0
$748$	− 4.42221i	− 0.161692i
$749$	0	0
$750$	0	0
$751$	6.02776	0.219956	0.109978	−	0.993934i	$-0.464922\pi$
$751$	6.02776	0.219956	0.109978	+	0.993934i	$0.464922\pi$
$752$	17.2111i	0.627624i
$753$	0	0
$754$	6.78890	0.247237
$755$	0	0
$756$	0	0
$757$	− 31.2111i	− 1.13439i	−0.823585	−	0.567193i	$-0.808029\pi$
$757$	− 31.2111i	− 1.13439i	0.823585	−	0.567193i	$-0.191971\pi$
$758$	18.7527i	0.681130i
$759$	0	0
$760$	0	0
$761$	−53.4500	−1.93756	−0.968780	−	0.247923i	$-0.920252\pi$
$761$	−53.4500	−1.93756	−0.968780	+	0.247923i	$0.920252\pi$
$762$	0	0
$763$	− 32.2389i	− 1.16713i
$764$	4.97224	0.179889
$765$	0	0
$766$	−24.2750	−0.877092
$767$	− 5.21110i	− 0.188162i
$768$	0	0
$769$	−41.0000	−1.47850	−0.739249	−	0.673432i	$-0.764819\pi$
$769$	−41.0000	−1.47850	−0.739249	+	0.673432i	$0.764819\pi$
$770$	0	0
$771$	0	0
$772$	6.60555i	0.237739i
$773$	16.8167i	0.604853i	0.953173	+	0.302426i	$0.0977967\pi$
$773$	16.8167i	0.604853i	−0.953173	+	0.302426i	$0.902203\pi$
$774$	0	0
$775$	0	0
$776$	−24.0000	−0.861550
$777$	0	0
$778$	5.44996i	0.195391i
$779$	9.39445	0.336591
$780$	0	0
$781$	−38.0555	−1.36173
$782$	− 21.9083i	− 0.783440i
$783$	0	0
$784$	46.9361	1.67629
$785$	0	0
$786$	0	0
$787$	2.97224i	0.105949i	0.998596	+	0.0529745i	$0.0168702\pi$
$787$	2.97224i	0.105949i	−0.998596	+	0.0529745i	$0.983130\pi$
$788$	0.358288i	0.0127635i
$789$	0	0
$790$	0	0
$791$	−3.63331	−0.129186
$792$	0	0
$793$	− 6.18335i	− 0.219577i
$794$	16.4222	0.582802
$795$	0	0
$796$	−4.00000	−0.141776
$797$	37.6611i	1.33402i	0.745047	+	0.667012i	$0.232427\pi$
$797$	37.6611i	1.33402i	−0.745047	+	0.667012i	$0.767573\pi$
$798$	0	0
$799$	29.2111	1.03341
$800$	0	0
$801$	0	0
$802$	39.0833i	1.38008i
$803$	− 14.0555i	− 0.496008i
$804$	0	0
$805$	0	0
$806$	−1.26662	−0.0446146
$807$	0	0
$808$	− 36.0000i	− 1.26648i
$809$	−50.6056	−1.77920	−0.889598	−	0.456744i	$-0.849016\pi$
$809$	−50.6056	−1.77920	−0.889598	+	0.456744i	$0.849016\pi$
$810$	0	0
$811$	42.4222	1.48965	0.744823	−	0.667263i	$-0.232534\pi$
$811$	42.4222	1.48965	0.744823	+	0.667263i	$0.232534\pi$
$812$	12.0000i	0.421117i
$813$	0	0
$814$	6.78890	0.237951
$815$	0	0
$816$	0	0
$817$	− 23.8167i	− 0.833239i
$818$	6.51388i	0.227752i
$819$	0	0
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	3.81665i	0.133040i	0.997785	+	0.0665201i	$0.0211897\pi$
$823$	3.81665i	0.133040i	−0.997785	+	0.0665201i	$0.978810\pi$
$824$	−12.0000	−0.418040
$825$	0	0
$826$	−51.6333	−1.79655
$827$	33.7889i	1.17496i	0.809240	+	0.587478i	$0.199879\pi$
$827$	33.7889i	1.17496i	−0.809240	+	0.587478i	$0.800121\pi$
$828$	0	0
$829$	27.2111	0.945081	0.472540	−	0.881309i	$-0.343337\pi$
$829$	27.2111	0.945081	0.472540	+	0.881309i	$0.343337\pi$
$830$	0	0
$831$	0	0
$832$	5.33894i	0.185094i
$833$	− 79.6611i	− 2.76009i
$834$	0	0
$835$	0	0
$836$	−2.84441	−0.0983760
$837$	0	0
$838$	− 16.9722i	− 0.586296i
$839$	−18.7889	−0.648665	−0.324332	−	0.945943i	$-0.605140\pi$
$839$	−18.7889	−0.648665	−0.324332	+	0.945943i	$0.605140\pi$
$840$	0	0
$841$	45.0555	1.55364
$842$	30.5139i	1.05158i
$843$	0	0
$844$	3.88057	0.133575
$845$	0	0
$846$	0	0
$847$	19.3944i	0.666401i
$848$	18.5139i	0.635769i
$849$	0	0
$850$	0	0
$851$	−6.00000	−0.205677
$852$	0	0
$853$	14.7889i	0.506362i	0.967419	+	0.253181i	$0.0814769\pi$
$853$	14.7889i	0.506362i	−0.967419	+	0.253181i	$0.918523\pi$
$854$	−61.2666	−2.09650
$855$	0	0
$856$	0	0
$857$	45.2389i	1.54533i	0.634814	+	0.772665i	$0.281077\pi$
$857$	45.2389i	1.54533i	−0.634814	+	0.772665i	$0.718923\pi$
$858$	0	0
$859$	−6.02776	−0.205664	−0.102832	−	0.994699i	$-0.532790\pi$
$859$	−6.02776	−0.205664	−0.102832	+	0.994699i	$0.532790\pi$
$860$	0	0
$861$	0	0
$862$	− 33.6333i	− 1.14556i
$863$	15.7889i	0.537460i	0.963216	+	0.268730i	$0.0866040\pi$
$863$	15.7889i	0.537460i	−0.963216	+	0.268730i	$0.913396\pi$
$864$	0	0
$865$	0	0
$866$	−36.7889	−1.25014
$867$	0	0
$868$	− 2.23886i	− 0.0759918i
$869$	−11.4500	−0.388413
$870$	0	0
$871$	9.21110	0.312106
$872$	− 21.0000i	− 0.711150i
$873$	0	0
$874$	−14.0917	−0.476658
$875$	0	0
$876$	0	0
$877$	56.6611i	1.91331i	0.291227	+	0.956654i	$0.405937\pi$
$877$	56.6611i	1.91331i	−0.291227	+	0.956654i	$0.594063\pi$
$878$	26.5694i	0.896674i
$879$	0	0
$880$	0	0
$881$	−32.6056	−1.09851	−0.549254	−	0.835655i	$-0.685088\pi$
$881$	−32.6056	−1.09851	−0.549254	+	0.835655i	$0.685088\pi$
$882$	0	0
$883$	− 5.81665i	− 0.195746i	−0.995199	−	0.0978730i	$-0.968796\pi$
$883$	− 5.81665i	− 0.195746i	0.995199	−	0.0978730i	$-0.0312039\pi$
$884$	1.02776	0.0345672
$885$	0	0
$886$	24.2750	0.815535
$887$	− 12.6333i	− 0.424185i	−0.977250	−	0.212092i	$-0.931972\pi$
$887$	− 12.6333i	− 0.424185i	0.977250	−	0.212092i	$-0.0680278\pi$
$888$	0	0
$889$	22.0555	0.739718
$890$	0	0
$891$	0	0
$892$	− 3.02776i	− 0.101377i
$893$	− 18.7889i	− 0.628746i
$894$	0	0
$895$	0	0
$896$	37.2666	1.24499
$897$	0	0
$898$	− 15.9445i	− 0.532074i
$899$	−13.8167	−0.460811
$900$	0	0
$901$	31.4222	1.04683
$902$	8.84441i	0.294487i
$903$	0	0
$904$	−2.36669	−0.0787150
$905$	0	0
$906$	0	0
$907$	− 30.4222i	− 1.01015i	−0.863075	−	0.505076i	$-0.831464\pi$
$907$	− 30.4222i	− 1.01015i	0.863075	−	0.505076i	$-0.168536\pi$
$908$	7.93608i	0.263368i
$909$	0	0
$910$	0	0
$911$	31.0278	1.02800	0.513998	−	0.857792i	$-0.328164\pi$
$911$	31.0278	1.02800	0.513998	+	0.857792i	$0.328164\pi$
$912$	0	0
$913$	− 7.81665i	− 0.258693i
$914$	−1.57779	−0.0521888
$915$	0	0
$916$	1.88057	0.0621358
$917$	− 27.6333i	− 0.912532i
$918$	0	0
$919$	2.42221	0.0799012	0.0399506	−	0.999202i	$-0.487280\pi$
$919$	2.42221	0.0799012	0.0399506	+	0.999202i	$0.487280\pi$
$920$	0	0
$921$	0	0
$922$	28.1833i	0.928169i
$923$	− 8.84441i	− 0.291117i
$924$	0	0
$925$	0	0
$926$	−19.8167	−0.651216
$927$	0	0
$928$	14.6056i	0.479451i
$929$	−26.6056	−0.872900	−0.436450	−	0.899729i	$-0.643764\pi$
$929$	−26.6056	−0.872900	−0.436450	+	0.899729i	$0.643764\pi$
$930$	0	0
$931$	−51.2389	−1.67929
$932$	− 5.44996i	− 0.178519i
$933$	0	0
$934$	2.88057	0.0942551
$935$	0	0
$936$	0	0
$937$	− 26.7889i	− 0.875155i	−0.899181	−	0.437578i	$-0.855837\pi$
$937$	− 26.7889i	− 0.875155i	0.899181	−	0.437578i	$-0.144163\pi$
$938$	− 91.2666i	− 2.97996i
$939$	0	0
$940$	0	0
$941$	28.4222	0.926537	0.463269	−	0.886218i	$-0.346676\pi$
$941$	28.4222	0.926537	0.463269	+	0.886218i	$0.346676\pi$
$942$	0	0
$943$	− 7.81665i	− 0.254545i
$944$	−28.4222	−0.925064
$945$	0	0
$946$	22.4222	0.729009
$947$	− 39.0000i	− 1.26733i	−0.773608	−	0.633665i	$-0.781550\pi$
$947$	− 39.0000i	− 1.26733i	0.773608	−	0.633665i	$-0.218450\pi$
$948$	0	0
$949$	3.26662	0.106039
$950$	0	0
$951$	0	0
$952$	− 77.4500i	− 2.51017i
$953$	26.8444i	0.869576i	0.900533	+	0.434788i	$0.143177\pi$
$953$	26.8444i	0.869576i	−0.900533	+	0.434788i	$0.856823\pi$
$954$	0	0
$955$	0	0
$956$	0.238859	0.00772525
$957$	0	0
$958$	− 21.0833i	− 0.681170i
$959$	22.1833	0.716338
$960$	0	0
$961$	−28.4222	−0.916845
$962$	1.57779i	0.0508701i
$963$	0	0
$964$	8.54163	0.275108
$965$	0	0
$966$	0	0
$967$	− 50.0000i	− 1.60789i	−0.594703	−	0.803946i	$-0.702730\pi$
$967$	− 50.0000i	− 1.60789i	0.594703	−	0.803946i	$-0.297270\pi$
$968$	12.6333i	0.406050i
$969$	0	0
$970$	0	0
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	0	0
$973$	− 18.4222i	− 0.590589i
$974$	10.6611	0.341603
$975$	0	0
$976$	−33.7250	−1.07951
$977$	0.788897i	0.0252391i	0.999920	+	0.0126195i	$0.00401703\pi$
$977$	0.788897i	0.0252391i	−0.999920	+	0.0126195i	$0.995983\pi$
$978$	0	0
$979$	20.3667	0.650922
$980$	0	0
$981$	0	0
$982$	− 16.6611i	− 0.531676i
$983$	0.633308i	0.0201994i	0.999949	+	0.0100997i	$0.00321489\pi$
$983$	0.633308i	0.0201994i	−0.999949	+	0.0100997i	$0.996785\pi$
$984$	0	0
$985$	0	0
$986$	−62.8444	−2.00137
$987$	0	0
$988$	− 0.661064i	− 0.0210312i
$989$	−19.8167	−0.630133
$990$	0	0
$991$	−50.8167	−1.61424	−0.807122	−	0.590385i	$-0.798976\pi$
$991$	−50.8167	−1.61424	−0.807122	+	0.590385i	$0.798976\pi$
$992$	− 2.72498i	− 0.0865182i
$993$	0	0
$994$	−87.6333	−2.77956
$995$	0	0
$996$	0	0
$997$	− 53.8722i	− 1.70615i	−0.521789	−	0.853074i	$-0.674735\pi$
$997$	− 53.8722i	− 1.70615i	0.521789	−	0.853074i	$-0.325265\pi$
$998$	− 35.9638i	− 1.13842i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.2.b.i.649.2		4
3.2	odd	2		675.2.b.h.649.3		4
5.2	odd	4		135.2.a.c.1.2	✓	2
5.3	odd	4		675.2.a.p.1.1		2
5.4	even	2	inner	675.2.b.i.649.3		4
15.2	even	4		135.2.a.d.1.1	yes	2
15.8	even	4		675.2.a.k.1.2		2
15.14	odd	2		675.2.b.h.649.2		4
20.7	even	4		2160.2.a.ba.1.1		2
35.27	even	4		6615.2.a.p.1.2		2
40.27	even	4		8640.2.a.ck.1.1		2
40.37	odd	4		8640.2.a.cr.1.2		2
45.2	even	12		405.2.e.j.271.2		4
45.7	odd	12		405.2.e.k.271.1		4
45.22	odd	12		405.2.e.k.136.1		4
45.32	even	12		405.2.e.j.136.2		4
60.47	odd	4		2160.2.a.y.1.1		2
105.62	odd	4		6615.2.a.v.1.1		2
120.77	even	4		8640.2.a.df.1.2		2
120.107	odd	4		8640.2.a.cy.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.2.a.c.1.2	✓	2	5.2	odd	4
135.2.a.d.1.1	yes	2	15.2	even	4
405.2.e.j.136.2		4	45.32	even	12
405.2.e.j.271.2		4	45.2	even	12
405.2.e.k.136.1		4	45.22	odd	12
405.2.e.k.271.1		4	45.7	odd	12
675.2.a.k.1.2		2	15.8	even	4
675.2.a.p.1.1		2	5.3	odd	4
675.2.b.h.649.2		4	15.14	odd	2
675.2.b.h.649.3		4	3.2	odd	2
675.2.b.i.649.2		4	1.1	even	1	trivial
675.2.b.i.649.3		4	5.4	even	2	inner
2160.2.a.y.1.1		2	60.47	odd	4
2160.2.a.ba.1.1		2	20.7	even	4
6615.2.a.p.1.2		2	35.27	even	4
6615.2.a.v.1.1		2	105.62	odd	4
8640.2.a.ck.1.1		2	40.27	even	4
8640.2.a.cr.1.2		2	40.37	odd	4
8640.2.a.cy.1.1		2	120.107	odd	4
8640.2.a.df.1.2		2	120.77	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.30278i q^{2} +0.302776 q^{4} -4.60555i q^{7} -3.00000i q^{8} +O(q^{10})\)	\(q-1.30278i q^{2} +0.302776 q^{4} -4.60555i q^{7} -3.00000i q^{8} -2.60555 q^{11} -0.605551i q^{13} -6.00000 q^{14} -3.30278 q^{16} +5.60555i q^{17} +3.60555 q^{19} +3.39445i q^{22} -3.00000i q^{23} -0.788897 q^{26} -1.39445i q^{28} -8.60555 q^{29} +1.60555 q^{31} -1.69722i q^{32} +7.30278 q^{34} -2.00000i q^{37} -4.69722i q^{38} +2.60555 q^{41} -6.60555i q^{43} -0.788897 q^{44} -3.90833 q^{46} -5.21110i q^{47} -14.2111 q^{49} -0.183346i q^{52} -5.60555i q^{53} -13.8167 q^{56} +11.2111i q^{58} +8.60555 q^{59} +10.2111 q^{61} -2.09167i q^{62} -8.81665 q^{64} +15.2111i q^{67} +1.69722i q^{68} +14.6056 q^{71} +5.39445i q^{73} -2.60555 q^{74} +1.09167 q^{76} +12.0000i q^{77} +4.39445 q^{79} -3.39445i q^{82} +3.00000i q^{83} -8.60555 q^{86} +7.81665i q^{88} -7.81665 q^{89} -2.78890 q^{91} -0.908327i q^{92} -6.78890 q^{94} -8.00000i q^{97} +18.5139i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{4}+O(q^{10}) \) 4 * q - 6 * q^4	\( 4 q - 6 q^{4} + 4 q^{11} - 24 q^{14} - 6 q^{16} - 32 q^{26} - 20 q^{29} - 8 q^{31} + 22 q^{34} - 4 q^{41} - 32 q^{44} + 6 q^{46} - 28 q^{49} - 12 q^{56} + 20 q^{59} + 12 q^{61} + 8 q^{64} + 44 q^{71} + 4 q^{74} + 26 q^{76} + 32 q^{79} - 20 q^{86} + 12 q^{89} - 40 q^{91} - 56 q^{94}+O(q^{100}) \) 4 * q - 6 * q^4 + 4 * q^11 - 24 * q^14 - 6 * q^16 - 32 * q^26 - 20 * q^29 - 8 * q^31 + 22 * q^34 - 4 * q^41 - 32 * q^44 + 6 * q^46 - 28 * q^49 - 12 * q^56 + 20 * q^59 + 12 * q^61 + 8 * q^64 + 44 * q^71 + 4 * q^74 + 26 * q^76 + 32 * q^79 - 20 * q^86 + 12 * q^89 - 40 * q^91 - 56 * q^94

Introduction

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