LMFDB - Embedded newform 675.2.b.h.649.1

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			649.1
Root			$-2.30278i$ of defining polynomial
Character	$\chi$	$=$	675.649
Dual form			675.2.b.h.649.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30278i q^{2} -3.30278 q^{4} +2.60555i q^{7} +3.00000i q^{8} +O(q^{10})$	$q-2.30278i q^{2} -3.30278 q^{4} +2.60555i q^{7} +3.00000i q^{8} -4.60555 q^{11} +6.60555i q^{13} +6.00000 q^{14} +0.302776 q^{16} +1.60555i q^{17} -3.60555 q^{19} +10.6056i q^{22} +3.00000i q^{23} +15.2111 q^{26} -8.60555i q^{28} +1.39445 q^{29} -5.60555 q^{31} +5.30278i q^{32} +3.69722 q^{34} -2.00000i q^{37} +8.30278i q^{38} +4.60555 q^{41} +0.605551i q^{43} +15.2111 q^{44} +6.90833 q^{46} -9.21110i q^{47} +0.211103 q^{49} -21.8167i q^{52} -1.60555i q^{53} -7.81665 q^{56} -3.21110i q^{58} -1.39445 q^{59} -4.21110 q^{61} +12.9083i q^{62} +12.8167 q^{64} +0.788897i q^{67} -5.30278i q^{68} -7.39445 q^{71} +12.6056i q^{73} -4.60555 q^{74} +11.9083 q^{76} -12.0000i q^{77} +11.6056 q^{79} -10.6056i q^{82} -3.00000i q^{83} +1.39445 q^{86} -13.8167i q^{88} -13.8167 q^{89} -17.2111 q^{91} -9.90833i q^{92} -21.2111 q^{94} -8.00000i q^{97} -0.486122i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 2.30278i	− 1.62831i	−0.580649	−	0.814154i	$-0.697201\pi$
$2$	− 2.30278i	− 1.62831i	0.580649	−	0.814154i	$-0.302799\pi$
$3$	0	0
$3$	0	0
$4$	−3.30278	−1.65139
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.60555i	0.984806i	0.870367	+	0.492403i	$0.163881\pi$
$7$	2.60555i	0.984806i	−0.870367	+	0.492403i	$0.836119\pi$
$8$	3.00000i	1.06066i
$9$	0	0
$10$	0	0
$11$	−4.60555	−1.38863	−0.694313	−	0.719673i	$-0.744292\pi$
$11$	−4.60555	−1.38863	−0.694313	+	0.719673i	$0.744292\pi$
$12$	0	0
$13$	6.60555i	1.83205i	0.401121	+	0.916025i	$0.368621\pi$
$13$	6.60555i	1.83205i	−0.401121	+	0.916025i	$0.631379\pi$
$14$	6.00000	1.60357
$15$	0	0
$16$	0.302776	0.0756939
$17$	1.60555i	0.389403i	0.980863	+	0.194702i	$0.0623739\pi$
$17$	1.60555i	0.389403i	−0.980863	+	0.194702i	$0.937626\pi$
$18$	0	0
$19$	−3.60555	−0.827170	−0.413585	−	0.910465i	$-0.635724\pi$
$19$	−3.60555	−0.827170	−0.413585	+	0.910465i	$0.635724\pi$
$20$	0	0
$21$	0	0
$22$	10.6056i	2.26111i
$23$	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
$26$	15.2111	2.98314
$27$	0	0
$28$	− 8.60555i	− 1.62630i
$29$	1.39445	0.258943	0.129471	−	0.991583i	$-0.458672\pi$
$29$	1.39445	0.258943	0.129471	+	0.991583i	$0.458672\pi$
$30$	0	0
$31$	−5.60555	−1.00679	−0.503393	−	0.864057i	$-0.667915\pi$
$31$	−5.60555	−1.00679	−0.503393	+	0.864057i	$0.667915\pi$
$32$	5.30278i	0.937407i
$33$	0	0
$34$	3.69722	0.634069
$35$	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	$-0.0525685\pi$
$38$	8.30278i	1.34689i
$39$	0	0
$40$	0	0
$41$	4.60555	0.719266	0.359633	−	0.933094i	$-0.382902\pi$
$41$	4.60555	0.719266	0.359633	+	0.933094i	$0.382902\pi$
$42$	0	0
$43$	0.605551i	0.0923457i	0.998933	+	0.0461729i	$0.0147025\pi$
$43$	0.605551i	0.0923457i	−0.998933	+	0.0461729i	$0.985297\pi$
$44$	15.2111	2.29316
$45$	0	0
$46$	6.90833	1.01858
$47$	− 9.21110i	− 1.34358i	−0.740743	−	0.671789i	$-0.765526\pi$
$47$	− 9.21110i	− 1.34358i	0.740743	−	0.671789i	$-0.234474\pi$
$48$	0	0
$49$	0.211103	0.0301575
$50$	0	0
$51$	0	0
$52$	− 21.8167i	− 3.02543i
$53$	− 1.60555i	− 0.220539i	−0.993902	−	0.110270i	$-0.964829\pi$
$53$	− 1.60555i	− 0.220539i	0.993902	−	0.110270i	$-0.0351715\pi$
$54$	0	0
$55$	0	0
$56$	−7.81665	−1.04454
$57$	0	0
$58$	− 3.21110i	− 0.421638i
$59$	−1.39445	−0.181542	−0.0907709	−	0.995872i	$-0.528933\pi$
$59$	−1.39445	−0.181542	−0.0907709	+	0.995872i	$0.528933\pi$
$60$	0	0
$61$	−4.21110	−0.539176	−0.269588	−	0.962976i	$-0.586888\pi$
$61$	−4.21110	−0.539176	−0.269588	+	0.962976i	$0.586888\pi$
$62$	12.9083i	1.63936i
$63$	0	0
$64$	12.8167	1.60208
$65$	0	0
$66$	0	0
$67$	0.788897i	0.0963792i	0.998838	+	0.0481896i	$0.0153452\pi$
$67$	0.788897i	0.0963792i	−0.998838	+	0.0481896i	$0.984655\pi$
$68$	− 5.30278i	− 0.643056i
$69$	0	0
$70$	0	0
$71$	−7.39445	−0.877560	−0.438780	−	0.898595i	$-0.644589\pi$
$71$	−7.39445	−0.877560	−0.438780	+	0.898595i	$0.644589\pi$
$72$	0	0
$73$	12.6056i	1.47537i	0.675146	+	0.737684i	$0.264081\pi$
$73$	12.6056i	1.47537i	−0.675146	+	0.737684i	$0.735919\pi$
$74$	−4.60555	−0.535384
$75$	0	0
$76$	11.9083	1.36598
$77$	− 12.0000i	− 1.36753i
$78$	0	0
$79$	11.6056	1.30573	0.652863	−	0.757476i	$-0.273568\pi$
$79$	11.6056	1.30573	0.652863	+	0.757476i	$0.273568\pi$
$80$	0	0
$81$	0	0
$82$	− 10.6056i	− 1.17119i
$83$	− 3.00000i	− 0.329293i	−0.986353	−	0.164646i	$-0.947352\pi$
$83$	− 3.00000i	− 0.329293i	0.986353	−	0.164646i	$-0.0526483\pi$
$84$	0	0
$85$	0	0
$86$	1.39445	0.150367
$87$	0	0
$88$	− 13.8167i	− 1.47286i
$89$	−13.8167	−1.46456	−0.732281	−	0.681002i	$-0.761544\pi$
$89$	−13.8167	−1.46456	−0.732281	+	0.681002i	$0.761544\pi$
$90$	0	0
$91$	−17.2111	−1.80421
$92$	− 9.90833i	− 1.03301i
$93$	0	0
$94$	−21.2111	−2.18776
$95$	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	$-0.133125\pi$
$98$	− 0.486122i	− 0.0491057i
$99$	0	0
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	−19.8167	−1.94318
$105$	0	0
$106$	−3.69722	−0.359106
$107$	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
$112$	0.788897i	0.0745438i
$113$	15.2111i	1.43094i	0.698643	+	0.715470i	$0.253787\pi$
$113$	15.2111i	1.43094i	−0.698643	+	0.715470i	$0.746213\pi$
$114$	0	0
$115$	0	0
$116$	−4.60555	−0.427615
$117$	0	0
$118$	3.21110i	0.295606i
$119$	−4.18335	−0.383487
$120$	0	0
$121$	10.2111	0.928282
$122$	9.69722i	0.877945i
$123$	0	0
$124$	18.5139	1.66260
$125$	0	0
$126$	0	0
$127$	19.2111i	1.70471i	0.522964	+	0.852355i	$0.324826\pi$
$127$	19.2111i	1.70471i	−0.522964	+	0.852355i	$0.675174\pi$
$128$	− 18.9083i	− 1.67128i
$129$	0	0
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	− 9.39445i	− 0.814602i
$134$	1.81665	0.156935
$135$	0	0
$136$	−4.81665	−0.413025
$137$	16.8167i	1.43674i	0.695659	+	0.718372i	$0.255112\pi$
$137$	16.8167i	1.43674i	−0.695659	+	0.718372i	$0.744888\pi$
$138$	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
$142$	17.0278i	1.42894i
$143$	− 30.4222i	− 2.54403i
$144$	0	0
$145$	0	0
$146$	29.0278	2.40235
$147$	0	0
$148$	6.60555i	0.542973i
$149$	23.0278	1.88651	0.943254	−	0.332073i	$-0.107748\pi$
$149$	23.0278	1.88651	0.943254	+	0.332073i	$0.107748\pi$
$150$	0	0
$151$	14.4222	1.17366	0.586831	−	0.809709i	$-0.300375\pi$
$151$	14.4222	1.17366	0.586831	+	0.809709i	$0.300375\pi$
$152$	− 10.8167i	− 0.877346i
$153$	0	0
$154$	−27.6333	−2.22676
$155$	0	0
$156$	0	0
$157$	17.8167i	1.42192i	0.703231	+	0.710962i	$0.251740\pi$
$157$	17.8167i	1.42192i	−0.703231	+	0.710962i	$0.748260\pi$
$158$	− 26.7250i	− 2.12613i
$159$	0	0
$160$	0	0
$161$	−7.81665	−0.616039
$162$	0	0
$163$	2.00000i	0.156652i	0.996928	+	0.0783260i	$0.0249575\pi$
$163$	2.00000i	0.156652i	−0.996928	+	0.0783260i	$0.975042\pi$
$164$	−15.2111	−1.18779
$165$	0	0
$166$	−6.90833	−0.536190
$167$	− 3.00000i	− 0.232147i	−0.993241	−	0.116073i	$-0.962969\pi$
$167$	− 3.00000i	− 0.232147i	0.993241	−	0.116073i	$-0.0370308\pi$
$168$	0	0
$169$	−30.6333	−2.35641
$170$	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
$176$	−1.39445	−0.105111
$177$	0	0
$178$	31.8167i	2.38476i
$179$	−21.2111	−1.58539	−0.792696	−	0.609617i	$-0.791323\pi$
$179$	−21.2111	−1.58539	−0.792696	+	0.609617i	$0.791323\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	39.6333i	2.93782i
$183$	0	0
$184$	−9.00000	−0.663489
$185$	0	0
$186$	0	0
$187$	− 7.39445i	− 0.540736i
$188$	30.4222i	2.21877i
$189$	0	0
$190$	0	0
$191$	12.4222	0.898839	0.449420	−	0.893321i	$-0.351631\pi$
$191$	12.4222	0.898839	0.449420	+	0.893321i	$0.351631\pi$
$192$	0	0
$193$	0.183346i	0.0131975i	0.999978	+	0.00659877i	$0.00210047\pi$
$193$	0.183346i	0.0131975i	−0.999978	+	0.00659877i	$0.997900\pi$
$194$	−18.4222	−1.32264
$195$	0	0
$196$	−0.697224	−0.0498017
$197$	− 22.8167i	− 1.62562i	−0.582529	−	0.812810i	$-0.697937\pi$
$197$	− 22.8167i	− 1.62562i	0.582529	−	0.812810i	$-0.302063\pi$
$198$	0	0
$199$	1.21110	0.0858528	0.0429264	−	0.999078i	$-0.486332\pi$
$199$	1.21110	0.0858528	0.0429264	+	0.999078i	$0.486332\pi$
$200$	0	0
$201$	0	0
$202$	27.6333i	1.94427i
$203$	3.63331i	0.255008i
$204$	0	0
$205$	0	0
$206$	−9.21110	−0.641768
$207$	0	0
$208$	2.00000i	0.138675i
$209$	16.6056	1.14863
$210$	0	0
$211$	−8.81665	−0.606963	−0.303482	−	0.952837i	$-0.598149\pi$
$211$	−8.81665	−0.606963	−0.303482	+	0.952837i	$0.598149\pi$
$212$	5.30278i	0.364196i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 14.6056i	− 0.991489i
$218$	− 16.1194i	− 1.09175i
$219$	0	0
$220$	0	0
$221$	−10.6056	−0.713407
$222$	0	0
$223$	− 10.0000i	− 0.669650i	−0.942280	−	0.334825i	$-0.891323\pi$
$223$	− 10.0000i	− 0.669650i	0.942280	−	0.334825i	$-0.108677\pi$
$224$	−13.8167	−0.923164
$225$	0	0
$226$	35.0278	2.33001
$227$	− 11.7889i	− 0.782457i	−0.920294	−	0.391228i	$-0.872050\pi$
$227$	− 11.7889i	− 0.782457i	0.920294	−	0.391228i	$-0.127950\pi$
$228$	0	0
$229$	−8.21110	−0.542605	−0.271302	−	0.962494i	$-0.587454\pi$
$229$	−8.21110	−0.542605	−0.271302	+	0.962494i	$0.587454\pi$
$230$	0	0
$231$	0	0
$232$	4.18335i	0.274650i
$233$	18.0000i	1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$	18.0000i	1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$	0	0
$235$	0	0
$236$	4.60555	0.299796
$237$	0	0
$238$	9.63331i	0.624435i
$239$	−15.2111	−0.983924	−0.491962	−	0.870617i	$-0.663720\pi$
$239$	−15.2111	−0.983924	−0.491962	+	0.870617i	$0.663720\pi$
$240$	0	0
$241$	13.7889	0.888221	0.444110	−	0.895972i	$-0.353520\pi$
$241$	13.7889	0.888221	0.444110	+	0.895972i	$0.353520\pi$
$242$	− 23.5139i	− 1.51153i
$243$	0	0
$244$	13.9083	0.890389
$245$	0	0
$246$	0	0
$247$	− 23.8167i	− 1.51542i
$248$	− 16.8167i	− 1.06786i
$249$	0	0
$250$	0	0
$251$	−27.6333	−1.74420	−0.872099	−	0.489329i	$-0.837242\pi$
$251$	−27.6333	−1.74420	−0.872099	+	0.489329i	$0.837242\pi$
$252$	0	0
$253$	− 13.8167i	− 0.868646i
$254$	44.2389	2.77579
$255$	0	0
$256$	−17.9083	−1.11927
$257$	1.18335i	0.0738151i	0.999319	+	0.0369076i	$0.0117507\pi$
$257$	1.18335i	0.0738151i	−0.999319	+	0.0369076i	$0.988249\pi$
$258$	0	0
$259$	5.21110	0.323802
$260$	0	0
$261$	0	0
$262$	13.8167i	0.853596i
$263$	2.78890i	0.171971i	0.996296	+	0.0859854i	$0.0274038\pi$
$263$	2.78890i	0.171971i	−0.996296	+	0.0859854i	$0.972596\pi$
$264$	0	0
$265$	0	0
$266$	−21.6333	−1.32642
$267$	0	0
$268$	− 2.60555i	− 0.159159i
$269$	−3.21110	−0.195784	−0.0978922	−	0.995197i	$-0.531210\pi$
$269$	−3.21110	−0.195784	−0.0978922	+	0.995197i	$0.531210\pi$
$270$	0	0
$271$	31.2389	1.89763	0.948813	−	0.315839i	$-0.102286\pi$
$271$	31.2389	1.89763	0.948813	+	0.315839i	$0.102286\pi$
$272$	0.486122i	0.0294755i
$273$	0	0
$274$	38.7250	2.33946
$275$	0	0
$276$	0	0
$277$	− 7.02776i	− 0.422257i	−0.977458	−	0.211128i	$-0.932286\pi$
$277$	− 7.02776i	− 0.422257i	0.977458	−	0.211128i	$-0.0677138\pi$
$278$	− 9.21110i	− 0.552445i
$279$	0	0
$280$	0	0
$281$	19.8167	1.18216	0.591081	−	0.806612i	$-0.298701\pi$
$281$	19.8167	1.18216	0.591081	+	0.806612i	$0.298701\pi$
$282$	0	0
$283$	3.39445i	0.201779i	0.994898	+	0.100890i	$0.0321689\pi$
$283$	3.39445i	0.201779i	−0.994898	+	0.100890i	$0.967831\pi$
$284$	24.4222	1.44919
$285$	0	0
$286$	−70.0555	−4.14247
$287$	12.0000i	0.708338i
$288$	0	0
$289$	14.4222	0.848365
$290$	0	0
$291$	0	0
$292$	− 41.6333i	− 2.43641i
$293$	7.18335i	0.419656i	0.977738	+	0.209828i	$0.0672903\pi$
$293$	7.18335i	0.419656i	−0.977738	+	0.209828i	$0.932710\pi$
$294$	0	0
$295$	0	0
$296$	6.00000	0.348743
$297$	0	0
$298$	− 53.0278i	− 3.07182i
$299$	−19.8167	−1.14603
$300$	0	0
$301$	−1.57779	−0.0909426
$302$	− 33.2111i	− 1.91108i
$303$	0	0
$304$	−1.09167	−0.0626117
$305$	0	0
$306$	0	0
$307$	− 8.42221i	− 0.480681i	−0.970689	−	0.240340i	$-0.922741\pi$
$307$	− 8.42221i	− 0.480681i	0.970689	−	0.240340i	$-0.0772590\pi$
$308$	39.6333i	2.25832i
$309$	0	0
$310$	0	0
$311$	7.81665	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$311$	7.81665	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$312$	0	0
$313$	− 19.6333i	− 1.10974i	−0.831937	−	0.554870i	$-0.812768\pi$
$313$	− 19.6333i	− 1.10974i	0.831937	−	0.554870i	$-0.187232\pi$
$314$	41.0278	2.31533
$315$	0	0
$316$	−38.3305	−2.15626
$317$	− 7.60555i	− 0.427170i	−0.976924	−	0.213585i	$-0.931486\pi$
$317$	− 7.60555i	− 0.427170i	0.976924	−	0.213585i	$-0.0685141\pi$
$318$	0	0
$319$	−6.42221	−0.359574
$320$	0	0
$321$	0	0
$322$	18.0000i	1.00310i
$323$	− 5.78890i	− 0.322103i
$324$	0	0
$325$	0	0
$326$	4.60555	0.255078
$327$	0	0
$328$	13.8167i	0.762897i
$329$	24.0000	1.32316
$330$	0	0
$331$	29.2111	1.60559	0.802794	−	0.596257i	$-0.203346\pi$
$331$	29.2111	1.60559	0.802794	+	0.596257i	$0.203346\pi$
$332$	9.90833i	0.543790i
$333$	0	0
$334$	−6.90833	−0.378007
$335$	0	0
$336$	0	0
$337$	− 6.60555i	− 0.359827i	−0.983682	−	0.179914i	$-0.942418\pi$
$337$	− 6.60555i	− 0.359827i	0.983682	−	0.179914i	$-0.0575818\pi$
$338$	70.5416i	3.83696i
$339$	0	0
$340$	0	0
$341$	25.8167	1.39805
$342$	0	0
$343$	18.7889i	1.01451i
$344$	−1.81665	−0.0979474
$345$	0	0
$346$	24.9083	1.33908
$347$	30.4222i	1.63315i	0.577240	+	0.816575i	$0.304130\pi$
$347$	30.4222i	1.63315i	−0.577240	+	0.816575i	$0.695870\pi$
$348$	0	0
$349$	31.8444	1.70459	0.852296	−	0.523060i	$-0.175209\pi$
$349$	31.8444	1.70459	0.852296	+	0.523060i	$0.175209\pi$
$350$	0	0
$351$	0	0
$352$	− 24.4222i	− 1.30171i
$353$	− 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
$356$	45.6333	2.41856
$357$	0	0
$358$	48.8444i	2.58151i
$359$	−9.63331	−0.508427	−0.254213	−	0.967148i	$-0.581817\pi$
$359$	−9.63331	−0.508427	−0.254213	+	0.967148i	$0.581817\pi$
$360$	0	0
$361$	−6.00000	−0.315789
$362$	16.1194i	0.847218i
$363$	0	0
$364$	56.8444	2.97946
$365$	0	0
$366$	0	0
$367$	2.60555i	0.136009i	0.997685	+	0.0680043i	$0.0216632\pi$
$367$	2.60555i	0.136009i	−0.997685	+	0.0680043i	$0.978337\pi$
$368$	0.908327i	0.0473498i
$369$	0	0
$370$	0	0
$371$	4.18335	0.217189
$372$	0	0
$373$	− 25.2111i	− 1.30538i	−0.757624	−	0.652691i	$-0.773640\pi$
$373$	− 25.2111i	− 1.30538i	0.757624	−	0.652691i	$-0.226360\pi$
$374$	−17.0278	−0.880484
$375$	0	0
$376$	27.6333	1.42508
$377$	9.21110i	0.474396i
$378$	0	0
$379$	−21.6056	−1.10980	−0.554901	−	0.831916i	$-0.687244\pi$
$379$	−21.6056	−1.10980	−0.554901	+	0.831916i	$0.687244\pi$
$380$	0	0
$381$	0	0
$382$	− 28.6056i	− 1.46359i
$383$	− 24.6333i	− 1.25870i	−0.777121	−	0.629352i	$-0.783321\pi$
$383$	− 24.6333i	− 1.25870i	0.777121	−	0.629352i	$-0.216679\pi$
$384$	0	0
$385$	0	0
$386$	0.422205	0.0214897
$387$	0	0
$388$	26.4222i	1.34138i
$389$	25.8167	1.30896	0.654478	−	0.756081i	$-0.272888\pi$
$389$	25.8167	1.30896	0.654478	+	0.756081i	$0.272888\pi$
$390$	0	0
$391$	−4.81665	−0.243589
$392$	0.633308i	0.0319869i
$393$	0	0
$394$	−52.5416	−2.64701
$395$	0	0
$396$	0	0
$397$	5.39445i	0.270740i	0.990795	+	0.135370i	$0.0432222\pi$
$397$	5.39445i	0.270740i	−0.990795	+	0.135370i	$0.956778\pi$
$398$	− 2.78890i	− 0.139795i
$399$	0	0
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	− 37.0278i	− 1.84448i
$404$	39.6333	1.97183
$405$	0	0
$406$	8.36669	0.415232
$407$	9.21110i	0.456577i
$408$	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
$412$	13.2111i	0.650864i
$413$	− 3.63331i	− 0.178783i
$414$	0	0
$415$	0	0
$416$	−35.0278	−1.71738
$417$	0	0
$418$	− 38.2389i	− 1.87032i
$419$	23.0278	1.12498	0.562490	−	0.826804i	$-0.309844\pi$
$419$	23.0278	1.12498	0.562490	+	0.826804i	$0.309844\pi$
$420$	0	0
$421$	5.42221	0.264262	0.132131	−	0.991232i	$-0.457818\pi$
$421$	5.42221	0.264262	0.132131	+	0.991232i	$0.457818\pi$
$422$	20.3028i	0.988324i
$423$	0	0
$424$	4.81665	0.233917
$425$	0	0
$426$	0	0
$427$	− 10.9722i	− 0.530984i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.18335	−0.201505	−0.100752	−	0.994912i	$-0.532125\pi$
$431$	−4.18335	−0.201505	−0.100752	+	0.994912i	$0.532125\pi$
$432$	0	0
$433$	22.2389i	1.06873i	0.845253	+	0.534366i	$0.179449\pi$
$433$	22.2389i	1.06873i	−0.845253	+	0.534366i	$0.820551\pi$
$434$	−33.6333	−1.61445
$435$	0	0
$436$	−23.1194	−1.10722
$437$	− 10.8167i	− 0.517431i
$438$	0	0
$439$	−27.6056	−1.31754	−0.658771	−	0.752344i	$-0.728923\pi$
$439$	−27.6056	−1.31754	−0.658771	+	0.752344i	$0.728923\pi$
$440$	0	0
$441$	0	0
$442$	24.4222i	1.16165i
$443$	24.6333i	1.17036i	0.810902	+	0.585182i	$0.198977\pi$
$443$	24.6333i	1.17036i	−0.810902	+	0.585182i	$0.801023\pi$
$444$	0	0
$445$	0	0
$446$	−23.0278	−1.09040
$447$	0	0
$448$	33.3944i	1.57774i
$449$	38.2389	1.80460	0.902302	−	0.431105i	$-0.141876\pi$
$449$	38.2389	1.80460	0.902302	+	0.431105i	$0.141876\pi$
$450$	0	0
$451$	−21.2111	−0.998792
$452$	− 50.2389i	− 2.36304i
$453$	0	0
$454$	−27.1472	−1.27408
$455$	0	0
$456$	0	0
$457$	13.2111i	0.617989i	0.951064	+	0.308995i	$0.0999925\pi$
$457$	13.2111i	0.617989i	−0.951064	+	0.308995i	$0.900007\pi$
$458$	18.9083i	0.883528i
$459$	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
$463$	− 0.788897i	− 0.0366632i	−0.999832	−	0.0183316i	$-0.994165\pi$
$463$	− 0.788897i	− 0.0366632i	0.999832	−	0.0183316i	$-0.00583545\pi$
$464$	0.422205	0.0196004
$465$	0	0
$466$	41.4500	1.92013
$467$	12.2111i	0.565062i	0.959258	+	0.282531i	$0.0911741\pi$
$467$	12.2111i	0.565062i	−0.959258	+	0.282531i	$0.908826\pi$
$468$	0	0
$469$	−2.05551	−0.0949148
$470$	0	0
$471$	0	0
$472$	− 4.18335i	− 0.192554i
$473$	− 2.78890i	− 0.128234i
$474$	0	0
$475$	0	0
$476$	13.8167	0.633285
$477$	0	0
$478$	35.0278i	1.60213i
$479$	−37.8167	−1.72789	−0.863944	−	0.503589i	$-0.832013\pi$
$479$	−37.8167	−1.72789	−0.863944	+	0.503589i	$0.832013\pi$
$480$	0	0
$481$	13.2111	0.602374
$482$	− 31.7527i	− 1.44630i
$483$	0	0
$484$	−33.7250	−1.53295
$485$	0	0
$486$	0	0
$487$	29.8167i	1.35112i	0.737304	+	0.675561i	$0.236098\pi$
$487$	29.8167i	1.35112i	−0.737304	+	0.675561i	$0.763902\pi$
$488$	− 12.6333i	− 0.571883i
$489$	0	0
$490$	0	0
$491$	−27.2111	−1.22802	−0.614010	−	0.789298i	$-0.710445\pi$
$491$	−27.2111	−1.22802	−0.614010	+	0.789298i	$0.710445\pi$
$492$	0	0
$493$	2.23886i	0.100833i
$494$	−54.8444	−2.46757
$495$	0	0
$496$	−1.69722	−0.0762076
$497$	− 19.2666i	− 0.864226i
$498$	0	0
$499$	20.3944	0.912981	0.456490	−	0.889728i	$-0.349106\pi$
$499$	20.3944	0.912981	0.456490	+	0.889728i	$0.349106\pi$
$500$	0	0
$501$	0	0
$502$	63.6333i	2.84009i
$503$	− 2.57779i	− 0.114938i	−0.998347	−	0.0574691i	$-0.981697\pi$
$503$	− 2.57779i	− 0.114938i	0.998347	−	0.0574691i	$-0.0183031\pi$
$504$	0	0
$505$	0	0
$506$	−31.8167	−1.41442
$507$	0	0
$508$	− 63.4500i	− 2.81514i
$509$	−24.8444	−1.10121	−0.550605	−	0.834766i	$-0.685603\pi$
$509$	−24.8444	−1.10121	−0.550605	+	0.834766i	$0.685603\pi$
$510$	0	0
$511$	−32.8444	−1.45295
$512$	3.42221i	0.151242i
$513$	0	0
$514$	2.72498	0.120194
$515$	0	0
$516$	0	0
$517$	42.4222i	1.86573i
$518$	− 12.0000i	− 0.527250i
$519$	0	0
$520$	0	0
$521$	28.6056	1.25323	0.626616	−	0.779328i	$-0.284439\pi$
$521$	28.6056	1.25323	0.626616	+	0.779328i	$0.284439\pi$
$522$	0	0
$523$	30.6056i	1.33829i	0.743133	+	0.669144i	$0.233339\pi$
$523$	30.6056i	1.33829i	−0.743133	+	0.669144i	$0.766661\pi$
$524$	19.8167	0.865695
$525$	0	0
$526$	6.42221	0.280021
$527$	− 9.00000i	− 0.392046i
$528$	0	0
$529$	14.0000	0.608696
$530$	0	0
$531$	0	0
$532$	31.0278i	1.34522i
$533$	30.4222i	1.31773i
$534$	0	0
$535$	0	0
$536$	−2.36669	−0.102226
$537$	0	0
$538$	7.39445i	0.318797i
$539$	−0.972244	−0.0418775
$540$	0	0
$541$	−40.4222	−1.73789	−0.868943	−	0.494912i	$-0.835200\pi$
$541$	−40.4222	−1.73789	−0.868943	+	0.494912i	$0.835200\pi$
$542$	− 71.9361i	− 3.08992i
$543$	0	0
$544$	−8.51388	−0.365030
$545$	0	0
$546$	0	0
$547$	− 0.605551i	− 0.0258915i	−0.999916	−	0.0129458i	$-0.995879\pi$
$547$	− 0.605551i	− 0.0258915i	0.999916	−	0.0129458i	$-0.00412088\pi$
$548$	− 55.5416i	− 2.37262i
$549$	0	0
$550$	0	0
$551$	−5.02776	−0.214190
$552$	0	0
$553$	30.2389i	1.28589i
$554$	−16.1833	−0.687564
$555$	0	0
$556$	−13.2111	−0.560276
$557$	9.63331i	0.408176i	0.978953	+	0.204088i	$0.0654229\pi$
$557$	9.63331i	0.408176i	−0.978953	+	0.204088i	$0.934577\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	− 45.6333i	− 1.92492i
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	0	0
$565$	0	0
$566$	7.81665	0.328558
$567$	0	0
$568$	− 22.1833i	− 0.930793i
$569$	16.1833	0.678441	0.339221	−	0.940707i	$-0.389837\pi$
$569$	16.1833	0.678441	0.339221	+	0.940707i	$0.389837\pi$
$570$	0	0
$571$	28.4500	1.19059	0.595297	−	0.803506i	$-0.297034\pi$
$571$	28.4500	1.19059	0.595297	+	0.803506i	$0.297034\pi$
$572$	100.478i	4.20118i
$573$	0	0
$574$	27.6333	1.15339
$575$	0	0
$576$	0	0
$577$	− 6.18335i	− 0.257416i	−0.991683	−	0.128708i	$-0.958917\pi$
$577$	− 6.18335i	− 0.257416i	0.991683	−	0.128708i	$-0.0410830\pi$
$578$	− 33.2111i	− 1.38140i
$579$	0	0
$580$	0	0
$581$	7.81665	0.324289
$582$	0	0
$583$	7.39445i	0.306247i
$584$	−37.8167	−1.56486
$585$	0	0
$586$	16.5416	0.683329
$587$	21.0000i	0.866763i	0.901211	+	0.433381i	$0.142680\pi$
$587$	21.0000i	0.866763i	−0.901211	+	0.433381i	$0.857320\pi$
$588$	0	0
$589$	20.2111	0.832784
$590$	0	0
$591$	0	0
$592$	− 0.605551i	− 0.0248880i
$593$	29.2389i	1.20070i	0.799739	+	0.600348i	$0.204971\pi$
$593$	29.2389i	1.20070i	−0.799739	+	0.600348i	$0.795029\pi$
$594$	0	0
$595$	0	0
$596$	−76.0555	−3.11536
$597$	0	0
$598$	45.6333i	1.86608i
$599$	−22.6056	−0.923638	−0.461819	−	0.886974i	$-0.652803\pi$
$599$	−22.6056	−0.923638	−0.461819	+	0.886974i	$0.652803\pi$
$600$	0	0
$601$	−10.6333	−0.433742	−0.216871	−	0.976200i	$-0.569585\pi$
$601$	−10.6333	−0.433742	−0.216871	+	0.976200i	$0.569585\pi$
$602$	3.63331i	0.148083i
$603$	0	0
$604$	−47.6333	−1.93817
$605$	0	0
$606$	0	0
$607$	− 24.6056i	− 0.998709i	−0.866398	−	0.499354i	$-0.833571\pi$
$607$	− 24.6056i	− 0.998709i	0.866398	−	0.499354i	$-0.166429\pi$
$608$	− 19.1194i	− 0.775395i
$609$	0	0
$610$	0	0
$611$	60.8444	2.46150
$612$	0	0
$613$	− 28.8444i	− 1.16501i	−0.812825	−	0.582507i	$-0.802072\pi$
$613$	− 28.8444i	− 1.16501i	0.812825	−	0.582507i	$-0.197928\pi$
$614$	−19.3944	−0.782696
$615$	0	0
$616$	36.0000	1.45048
$617$	38.4500i	1.54794i	0.633224	+	0.773969i	$0.281731\pi$
$617$	38.4500i	1.54794i	−0.633224	+	0.773969i	$0.718269\pi$
$618$	0	0
$619$	−35.6333	−1.43222	−0.716112	−	0.697986i	$-0.754080\pi$
$619$	−35.6333	−1.43222	−0.716112	+	0.697986i	$0.754080\pi$
$620$	0	0
$621$	0	0
$622$	− 18.0000i	− 0.721734i
$623$	− 36.0000i	− 1.44231i
$624$	0	0
$625$	0	0
$626$	−45.2111	−1.80700
$627$	0	0
$628$	− 58.8444i	− 2.34815i
$629$	3.21110	0.128035
$630$	0	0
$631$	−6.02776	−0.239961	−0.119981	−	0.992776i	$-0.538283\pi$
$631$	−6.02776	−0.239961	−0.119981	+	0.992776i	$0.538283\pi$
$632$	34.8167i	1.38493i
$633$	0	0
$634$	−17.5139	−0.695565
$635$	0	0
$636$	0	0
$637$	1.39445i	0.0552501i
$638$	14.7889i	0.585498i
$639$	0	0
$640$	0	0
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	0	0
$643$	13.0278i	0.513765i	0.966443	+	0.256882i	$0.0826953\pi$
$643$	13.0278i	0.513765i	−0.966443	+	0.256882i	$0.917305\pi$
$644$	25.8167	1.01732
$645$	0	0
$646$	−13.3305	−0.524483
$647$	23.7889i	0.935238i	0.883930	+	0.467619i	$0.154888\pi$
$647$	23.7889i	0.935238i	−0.883930	+	0.467619i	$0.845112\pi$
$648$	0	0
$649$	6.42221	0.252094
$650$	0	0
$651$	0	0
$652$	− 6.60555i	− 0.258693i
$653$	23.2389i	0.909407i	0.890643	+	0.454703i	$0.150255\pi$
$653$	23.2389i	0.909407i	−0.890643	+	0.454703i	$0.849745\pi$
$654$	0	0
$655$	0	0
$656$	1.39445	0.0544441
$657$	0	0
$658$	− 55.2666i	− 2.15452i
$659$	−13.3944	−0.521774	−0.260887	−	0.965369i	$-0.584015\pi$
$659$	−13.3944	−0.521774	−0.260887	+	0.965369i	$0.584015\pi$
$660$	0	0
$661$	−34.8444	−1.35529	−0.677645	−	0.735389i	$-0.736999\pi$
$661$	−34.8444	−1.35529	−0.677645	+	0.735389i	$0.736999\pi$
$662$	− 67.2666i	− 2.61439i
$663$	0	0
$664$	9.00000	0.349268
$665$	0	0
$666$	0	0
$667$	4.18335i	0.161980i
$668$	9.90833i	0.383365i
$669$	0	0
$670$	0	0
$671$	19.3944	0.748714
$672$	0	0
$673$	25.0278i	0.964749i	0.875965	+	0.482375i	$0.160226\pi$
$673$	25.0278i	0.964749i	−0.875965	+	0.482375i	$0.839774\pi$
$674$	−15.2111	−0.585910
$675$	0	0
$676$	101.175	3.89134
$677$	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
$679$	20.8444	0.799935
$680$	0	0
$681$	0	0
$682$	− 59.4500i	− 2.27646i
$683$	9.00000i	0.344375i	0.985064	+	0.172188i	$0.0550836\pi$
$683$	9.00000i	0.344375i	−0.985064	+	0.172188i	$0.944916\pi$
$684$	0	0
$685$	0	0
$686$	43.2666	1.65193
$687$	0	0
$688$	0.183346i	0.00699001i
$689$	10.6056	0.404039
$690$	0	0
$691$	9.60555	0.365412	0.182706	−	0.983168i	$-0.441514\pi$
$691$	9.60555	0.365412	0.182706	+	0.983168i	$0.441514\pi$
$692$	− 35.7250i	− 1.35806i
$693$	0	0
$694$	70.0555	2.65927
$695$	0	0
$696$	0	0
$697$	7.39445i	0.280085i
$698$	− 73.3305i	− 2.77560i
$699$	0	0
$700$	0	0
$701$	−11.5778	−0.437287	−0.218644	−	0.975805i	$-0.570163\pi$
$701$	−11.5778	−0.437287	−0.218644	+	0.975805i	$0.570163\pi$
$702$	0	0
$703$	7.21110i	0.271972i
$704$	−59.0278	−2.22469
$705$	0	0
$706$	−49.8167	−1.87487
$707$	− 31.2666i	− 1.17590i
$708$	0	0
$709$	22.8444	0.857940	0.428970	−	0.903319i	$-0.358877\pi$
$709$	22.8444	0.857940	0.428970	+	0.903319i	$0.358877\pi$
$710$	0	0
$711$	0	0
$712$	− 41.4500i	− 1.55340i
$713$	− 16.8167i	− 0.629789i
$714$	0	0
$715$	0	0
$716$	70.0555	2.61810
$717$	0	0
$718$	22.1833i	0.827875i
$719$	37.2666	1.38981	0.694905	−	0.719101i	$-0.255446\pi$
$719$	37.2666	1.38981	0.694905	+	0.719101i	$0.255446\pi$
$720$	0	0
$721$	10.4222	0.388143
$722$	13.8167i	0.514203i
$723$	0	0
$724$	23.1194	0.859227
$725$	0	0
$726$	0	0
$727$	− 35.6333i	− 1.32157i	−0.750577	−	0.660783i	$-0.770224\pi$
$727$	− 35.6333i	− 1.32157i	0.750577	−	0.660783i	$-0.229776\pi$
$728$	− 51.6333i	− 1.91366i
$729$	0	0
$730$	0	0
$731$	−0.972244	−0.0359597
$732$	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
$736$	−15.9083	−0.586389
$737$	− 3.63331i	− 0.133835i
$738$	0	0
$739$	6.02776	0.221735	0.110867	−	0.993835i	$-0.464637\pi$
$739$	6.02776	0.221735	0.110867	+	0.993835i	$0.464637\pi$
$740$	0	0
$741$	0	0
$742$	− 9.63331i	− 0.353650i
$743$	5.57779i	0.204629i	0.994752	+	0.102315i	$0.0326249\pi$
$743$	5.57779i	0.204629i	−0.994752	+	0.102315i	$0.967375\pi$
$744$	0	0
$745$	0	0
$746$	−58.0555	−2.12556
$747$	0	0
$748$	24.4222i	0.892964i
$749$	0	0
$750$	0	0
$751$	−30.0278	−1.09573	−0.547864	−	0.836567i	$-0.684559\pi$
$751$	−30.0278	−1.09573	−0.547864	+	0.836567i	$0.684559\pi$
$752$	− 2.78890i	− 0.101701i
$753$	0	0
$754$	21.2111	0.772463
$755$	0	0
$756$	0	0
$757$	− 16.7889i	− 0.610203i	−0.952320	−	0.305101i	$-0.901310\pi$
$757$	− 16.7889i	− 0.610203i	0.952320	−	0.305101i	$-0.0986904\pi$
$758$	49.7527i	1.80710i
$759$	0	0
$760$	0	0
$761$	−11.4500	−0.415061	−0.207530	−	0.978229i	$-0.566543\pi$
$761$	−11.4500	−0.415061	−0.207530	+	0.978229i	$0.566543\pi$
$762$	0	0
$763$	18.2389i	0.660291i
$764$	−41.0278	−1.48433
$765$	0	0
$766$	−56.7250	−2.04956
$767$	− 9.21110i	− 0.332594i
$768$	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
$772$	− 0.605551i	− 0.0217943i
$773$	4.81665i	0.173243i	0.996241	+	0.0866215i	$0.0276071\pi$
$773$	4.81665i	0.173243i	−0.996241	+	0.0866215i	$0.972393\pi$
$774$	0	0
$775$	0	0
$776$	24.0000	0.861550
$777$	0	0
$778$	− 59.4500i	− 2.13138i
$779$	−16.6056	−0.594956
$780$	0	0
$781$	34.0555	1.21860
$782$	11.0917i	0.396637i
$783$	0	0
$784$	0.0639167	0.00228274
$785$	0	0
$786$	0	0
$787$	39.0278i	1.39119i	0.718434	+	0.695595i	$0.244859\pi$
$787$	39.0278i	1.39119i	−0.718434	+	0.695595i	$0.755141\pi$
$788$	75.3583i	2.68453i
$789$	0	0
$790$	0	0
$791$	−39.6333	−1.40920
$792$	0	0
$793$	− 27.8167i	− 0.987798i
$794$	12.4222	0.440848
$795$	0	0
$796$	−4.00000	−0.141776
$797$	41.6611i	1.47571i	0.674959	+	0.737855i	$0.264161\pi$
$797$	41.6611i	1.47571i	−0.674959	+	0.737855i	$0.735839\pi$
$798$	0	0
$799$	14.7889	0.523194
$800$	0	0
$801$	0	0
$802$	− 69.0833i	− 2.43942i
$803$	− 58.0555i	− 2.04873i
$804$	0	0
$805$	0	0
$806$	−85.2666	−3.00339
$807$	0	0
$808$	− 36.0000i	− 1.26648i
$809$	43.3944	1.52567	0.762834	−	0.646595i	$-0.223807\pi$
$809$	43.3944	1.52567	0.762834	+	0.646595i	$0.223807\pi$
$810$	0	0
$811$	13.5778	0.476781	0.238390	−	0.971169i	$-0.423380\pi$
$811$	13.5778	0.476781	0.238390	+	0.971169i	$0.423380\pi$
$812$	− 12.0000i	− 0.421117i
$813$	0	0
$814$	21.2111	0.743449
$815$	0	0
$816$	0	0
$817$	− 2.18335i	− 0.0763856i
$818$	11.5139i	0.402573i
$819$	0	0
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	− 17.8167i	− 0.621050i	−0.950565	−	0.310525i	$-0.899495\pi$
$823$	− 17.8167i	− 0.621050i	0.950565	−	0.310525i	$-0.100505\pi$
$824$	12.0000	0.418040
$825$	0	0
$826$	−8.36669	−0.291114
$827$	− 48.2111i	− 1.67646i	−0.545314	−	0.838232i	$-0.683589\pi$
$827$	− 48.2111i	− 1.67646i	0.545314	−	0.838232i	$-0.316411\pi$
$828$	0	0
$829$	12.7889	0.444177	0.222088	−	0.975027i	$-0.428713\pi$
$829$	12.7889	0.444177	0.222088	+	0.975027i	$0.428713\pi$
$830$	0	0
$831$	0	0
$832$	84.6611i	2.93509i
$833$	0.338936i	0.0117434i
$834$	0	0
$835$	0	0
$836$	−54.8444	−1.89683
$837$	0	0
$838$	− 53.0278i	− 1.83181i
$839$	33.2111	1.14657	0.573287	−	0.819354i	$-0.305668\pi$
$839$	33.2111	1.14657	0.573287	+	0.819354i	$0.305668\pi$
$840$	0	0
$841$	−27.0555	−0.932949
$842$	− 12.4861i	− 0.430300i
$843$	0	0
$844$	29.1194	1.00233
$845$	0	0
$846$	0	0
$847$	26.6056i	0.914178i
$848$	− 0.486122i	− 0.0166935i
$849$	0	0
$850$	0	0
$851$	6.00000	0.205677
$852$	0	0
$853$	29.2111i	1.00017i	0.865977	+	0.500085i	$0.166698\pi$
$853$	29.2111i	1.00017i	−0.865977	+	0.500085i	$0.833302\pi$
$854$	−25.2666	−0.864606
$855$	0	0
$856$	0	0
$857$	5.23886i	0.178956i	0.995989	+	0.0894780i	$0.0285199\pi$
$857$	5.23886i	0.178956i	−0.995989	+	0.0894780i	$0.971480\pi$
$858$	0	0
$859$	30.0278	1.02453	0.512267	−	0.858826i	$-0.328806\pi$
$859$	30.0278	1.02453	0.512267	+	0.858826i	$0.328806\pi$
$860$	0	0
$861$	0	0
$862$	9.63331i	0.328112i
$863$	− 30.2111i	− 1.02840i	−0.857671	−	0.514199i	$-0.828089\pi$
$863$	− 30.2111i	− 1.02840i	0.857671	−	0.514199i	$-0.171911\pi$
$864$	0	0
$865$	0	0
$866$	51.2111	1.74022
$867$	0	0
$868$	48.2389i	1.63733i
$869$	−53.4500	−1.81317
$870$	0	0
$871$	−5.21110	−0.176571
$872$	21.0000i	0.711150i
$873$	0	0
$874$	−24.9083	−0.842537
$875$	0	0
$876$	0	0
$877$	− 22.6611i	− 0.765210i	−0.923912	−	0.382605i	$-0.875027\pi$
$877$	− 22.6611i	− 0.765210i	0.923912	−	0.382605i	$-0.124973\pi$
$878$	63.5694i	2.14536i
$879$	0	0
$880$	0	0
$881$	25.3944	0.855561	0.427780	−	0.903883i	$-0.359296\pi$
$881$	25.3944	0.855561	0.427780	+	0.903883i	$0.359296\pi$
$882$	0	0
$883$	15.8167i	0.532273i	0.963935	+	0.266136i	$0.0857472\pi$
$883$	15.8167i	0.532273i	−0.963935	+	0.266136i	$0.914253\pi$
$884$	35.0278	1.17811
$885$	0	0
$886$	56.7250	1.90571
$887$	− 30.6333i	− 1.02857i	−0.857621	−	0.514283i	$-0.828058\pi$
$887$	− 30.6333i	− 1.02857i	0.857621	−	0.514283i	$-0.171942\pi$
$888$	0	0
$889$	−50.0555	−1.67881
$890$	0	0
$891$	0	0
$892$	33.0278i	1.10585i
$893$	33.2111i	1.11137i
$894$	0	0
$895$	0	0
$896$	49.2666	1.64588
$897$	0	0
$898$	− 88.0555i	− 2.93845i
$899$	−7.81665	−0.260700
$900$	0	0
$901$	2.57779	0.0858788
$902$	48.8444i	1.62634i
$903$	0	0
$904$	−45.6333	−1.51774
$905$	0	0
$906$	0	0
$907$	− 1.57779i	− 0.0523898i	−0.999657	−	0.0261949i	$-0.991661\pi$
$907$	− 1.57779i	− 0.0523898i	0.999657	−	0.0261949i	$-0.00833905\pi$
$908$	38.9361i	1.29214i
$909$	0	0
$910$	0	0
$911$	5.02776	0.166577	0.0832885	−	0.996525i	$-0.473458\pi$
$911$	5.02776	0.166577	0.0832885	+	0.996525i	$0.473458\pi$
$912$	0	0
$913$	13.8167i	0.457265i
$914$	30.4222	1.00628
$915$	0	0
$916$	27.1194	0.896051
$917$	− 15.6333i	− 0.516257i
$918$	0	0
$919$	−26.4222	−0.871588	−0.435794	−	0.900046i	$-0.643532\pi$
$919$	−26.4222	−0.871588	−0.435794	+	0.900046i	$0.643532\pi$
$920$	0	0
$921$	0	0
$922$	49.8167i	1.64062i
$923$	− 48.8444i	− 1.60773i
$924$	0	0
$925$	0	0
$926$	−1.81665	−0.0596989
$927$	0	0
$928$	7.39445i	0.242735i
$929$	19.3944	0.636311	0.318156	−	0.948039i	$-0.396937\pi$
$929$	19.3944	0.636311	0.318156	+	0.948039i	$0.396937\pi$
$930$	0	0
$931$	−0.761141	−0.0249454
$932$	− 59.4500i	− 1.94735i
$933$	0	0
$934$	28.1194	0.920096
$935$	0	0
$936$	0	0
$937$	− 41.2111i	− 1.34631i	−0.739502	−	0.673154i	$-0.764939\pi$
$937$	− 41.2111i	− 1.34631i	0.739502	−	0.673154i	$-0.235061\pi$
$938$	4.73338i	0.154550i
$939$	0	0
$940$	0	0
$941$	0.422205	0.0137635	0.00688175	−	0.999976i	$-0.497809\pi$
$941$	0.422205	0.0137635	0.00688175	+	0.999976i	$0.497809\pi$
$942$	0	0
$943$	13.8167i	0.449932i
$944$	−0.422205	−0.0137416
$945$	0	0
$946$	−6.42221	−0.208804
$947$	39.0000i	1.26733i	0.773608	+	0.633665i	$0.218450\pi$
$947$	39.0000i	1.26733i	−0.773608	+	0.633665i	$0.781550\pi$
$948$	0	0
$949$	−83.2666	−2.70295
$950$	0	0
$951$	0	0
$952$	− 12.5500i	− 0.406749i
$953$	30.8444i	0.999148i	0.866271	+	0.499574i	$0.166510\pi$
$953$	30.8444i	0.999148i	−0.866271	+	0.499574i	$0.833490\pi$
$954$	0	0
$955$	0	0
$956$	50.2389	1.62484
$957$	0	0
$958$	87.0833i	2.81353i
$959$	−43.8167	−1.41491
$960$	0	0
$961$	0.422205	0.0136195
$962$	− 30.4222i	− 0.980851i
$963$	0	0
$964$	−45.5416	−1.46680
$965$	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	$-0.297270\pi$
$968$	30.6333i	0.984592i
$969$	0	0
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	0	0
$973$	10.4222i	0.334121i
$974$	68.6611	2.20004
$975$	0	0
$976$	−1.27502	−0.0408124
$977$	− 15.2111i	− 0.486646i	−0.969945	−	0.243323i	$-0.921762\pi$
$977$	− 15.2111i	− 0.486646i	0.969945	−	0.243323i	$-0.0782375\pi$
$978$	0	0
$979$	63.6333	2.03373
$980$	0	0
$981$	0	0
$982$	62.6611i	1.99959i
$983$	42.6333i	1.35979i	0.733309	+	0.679896i	$0.237975\pi$
$983$	42.6333i	1.35979i	−0.733309	+	0.679896i	$0.762025\pi$
$984$	0	0
$985$	0	0
$986$	5.15559	0.164187
$987$	0	0
$988$	78.6611i	2.50254i
$989$	−1.81665	−0.0577662
$990$	0	0
$991$	−29.1833	−0.927040	−0.463520	−	0.886087i	$-0.653414\pi$
$991$	−29.1833	−0.927040	−0.463520	+	0.886087i	$0.653414\pi$
$992$	− 29.7250i	− 0.943769i
$993$	0	0
$994$	−44.3667	−1.40723
$995$	0	0
$996$	0	0
$997$	39.8722i	1.26276i	0.775472	+	0.631382i	$0.217512\pi$
$997$	39.8722i	1.26276i	−0.775472	+	0.631382i	$0.782488\pi$
$998$	− 46.9638i	− 1.48661i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.2.b.h.649.1		4
3.2	odd	2		675.2.b.i.649.4		4
5.2	odd	4		135.2.a.d.1.2	yes	2
5.3	odd	4		675.2.a.k.1.1		2
5.4	even	2	inner	675.2.b.h.649.4		4
15.2	even	4		135.2.a.c.1.1	✓	2
15.8	even	4		675.2.a.p.1.2		2
15.14	odd	2		675.2.b.i.649.1		4
20.7	even	4		2160.2.a.y.1.2		2
35.27	even	4		6615.2.a.v.1.2		2
40.27	even	4		8640.2.a.cy.1.2		2
40.37	odd	4		8640.2.a.df.1.1		2
45.2	even	12		405.2.e.k.271.2		4
45.7	odd	12		405.2.e.j.271.1		4
45.22	odd	12		405.2.e.j.136.1		4
45.32	even	12		405.2.e.k.136.2		4
60.47	odd	4		2160.2.a.ba.1.2		2
105.62	odd	4		6615.2.a.p.1.1		2
120.77	even	4		8640.2.a.cr.1.1		2
120.107	odd	4		8640.2.a.ck.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.2.a.c.1.1	✓	2	15.2	even	4
135.2.a.d.1.2	yes	2	5.2	odd	4
405.2.e.j.136.1		4	45.22	odd	12
405.2.e.j.271.1		4	45.7	odd	12
405.2.e.k.136.2		4	45.32	even	12
405.2.e.k.271.2		4	45.2	even	12
675.2.a.k.1.1		2	5.3	odd	4
675.2.a.p.1.2		2	15.8	even	4
675.2.b.h.649.1		4	1.1	even	1	trivial
675.2.b.h.649.4		4	5.4	even	2	inner
675.2.b.i.649.1		4	15.14	odd	2
675.2.b.i.649.4		4	3.2	odd	2
2160.2.a.y.1.2		2	20.7	even	4
2160.2.a.ba.1.2		2	60.47	odd	4
6615.2.a.p.1.1		2	105.62	odd	4
6615.2.a.v.1.2		2	35.27	even	4
8640.2.a.ck.1.2		2	120.107	odd	4
8640.2.a.cr.1.1		2	120.77	even	4
8640.2.a.cy.1.2		2	40.27	even	4
8640.2.a.df.1.1		2	40.37	odd	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)

\(f(q)\)	\(=\)	\(q-2.30278i q^{2} -3.30278 q^{4} +2.60555i q^{7} +3.00000i q^{8} +O(q^{10})\)	\(q-2.30278i q^{2} -3.30278 q^{4} +2.60555i q^{7} +3.00000i q^{8} -4.60555 q^{11} +6.60555i q^{13} +6.00000 q^{14} +0.302776 q^{16} +1.60555i q^{17} -3.60555 q^{19} +10.6056i q^{22} +3.00000i q^{23} +15.2111 q^{26} -8.60555i q^{28} +1.39445 q^{29} -5.60555 q^{31} +5.30278i q^{32} +3.69722 q^{34} -2.00000i q^{37} +8.30278i q^{38} +4.60555 q^{41} +0.605551i q^{43} +15.2111 q^{44} +6.90833 q^{46} -9.21110i q^{47} +0.211103 q^{49} -21.8167i q^{52} -1.60555i q^{53} -7.81665 q^{56} -3.21110i q^{58} -1.39445 q^{59} -4.21110 q^{61} +12.9083i q^{62} +12.8167 q^{64} +0.788897i q^{67} -5.30278i q^{68} -7.39445 q^{71} +12.6056i q^{73} -4.60555 q^{74} +11.9083 q^{76} -12.0000i q^{77} +11.6056 q^{79} -10.6056i q^{82} -3.00000i q^{83} +1.39445 q^{86} -13.8167i q^{88} -13.8167 q^{89} -17.2111 q^{91} -9.90833i q^{92} -21.2111 q^{94} -8.00000i q^{97} -0.486122i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{4}+O(q^{10}) \) 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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