LMFDB - Embedded newform 2475.2.c.c.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(199,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$19.7629745003$
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			199.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2475.199
Dual form			2475.2.c.c.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{4} -3.00000i q^{7} +3.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} +1.00000 q^{4} -3.00000i q^{7} +3.00000i q^{8} -1.00000 q^{11} -2.00000i q^{13} +3.00000 q^{14} -1.00000 q^{16} +3.00000i q^{17} +1.00000 q^{19} -1.00000i q^{22} -1.00000i q^{23} +2.00000 q^{26} -3.00000i q^{28} +6.00000 q^{29} +4.00000 q^{31} +5.00000i q^{32} -3.00000 q^{34} +1.00000i q^{37} +1.00000i q^{38} +5.00000 q^{41} -4.00000i q^{43} -1.00000 q^{44} +1.00000 q^{46} +3.00000i q^{47} -2.00000 q^{49} -2.00000i q^{52} -10.0000i q^{53} +9.00000 q^{56} +6.00000i q^{58} +11.0000 q^{59} +14.0000 q^{61} +4.00000i q^{62} -7.00000 q^{64} +2.00000i q^{67} +3.00000i q^{68} +5.00000 q^{71} -2.00000i q^{73} -1.00000 q^{74} +1.00000 q^{76} +3.00000i q^{77} -5.00000 q^{79} +5.00000i q^{82} -8.00000i q^{83} +4.00000 q^{86} -3.00000i q^{88} -10.0000 q^{89} -6.00000 q^{91} -1.00000i q^{92} -3.00000 q^{94} -17.0000i q^{97} -2.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4}+O(q^{10}) $ 2 * q + 2 * q^4	$ 2 q + 2 q^{4} - 2 q^{11} + 6 q^{14} - 2 q^{16} + 2 q^{19} + 4 q^{26} + 12 q^{29} + 8 q^{31} - 6 q^{34} + 10 q^{41} - 2 q^{44} + 2 q^{46} - 4 q^{49} + 18 q^{56} + 22 q^{59} + 28 q^{61} - 14 q^{64} + 10 q^{71} - 2 q^{74} + 2 q^{76} - 10 q^{79} + 8 q^{86} - 20 q^{89} - 12 q^{91} - 6 q^{94}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^11 + 6 * q^14 - 2 * q^16 + 2 * q^19 + 4 * q^26 + 12 * q^29 + 8 * q^31 - 6 * q^34 + 10 * q^41 - 2 * q^44 + 2 * q^46 - 4 * q^49 + 18 * q^56 + 22 * q^59 + 28 * q^61 - 14 * q^64 + 10 * q^71 - 2 * q^74 + 2 * q^76 - 10 * q^79 + 8 * q^86 - 20 * q^89 - 12 * q^91 - 6 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$.

$n$	$551$	$2026$	$2377$
$\chi(n)$	$1$	$1$	$-1$

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	0.935414	+	0.353553i	$0.115027\pi$
$2$	1.00000i	0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.00000i	− 1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	− 3.00000i	− 1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	3.00000i	1.06066i
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	3.00000	0.801784
$15$	0	0
$16$	−1.00000	−0.250000
$17$	3.00000i	0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$	3.00000i	0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	− 1.00000i	− 0.213201i
$23$	− 1.00000i	− 0.208514i	−0.994550	−	0.104257i	$-0.966753\pi$
$23$	− 1.00000i	− 0.208514i	0.994550	−	0.104257i	$-0.0332465\pi$
$24$	0	0
$25$	0	0
$26$	2.00000	0.392232
$27$	0	0
$28$	− 3.00000i	− 0.566947i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	5.00000i	0.883883i
$33$	0	0
$34$	−3.00000	−0.514496
$35$	0	0
$36$	0	0
$37$	1.00000i	0.164399i	0.996616	+	0.0821995i	$0.0261945\pi$
$37$	1.00000i	0.164399i	−0.996616	+	0.0821995i	$0.973806\pi$
$38$	1.00000i	0.162221i
$39$	0	0
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	1.00000	0.147442
$47$	3.00000i	0.437595i	0.975770	+	0.218797i	$0.0702134\pi$
$47$	3.00000i	0.437595i	−0.975770	+	0.218797i	$0.929787\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	0	0
$52$	− 2.00000i	− 0.277350i
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	0	0
$55$	0	0
$56$	9.00000	1.20268
$57$	0	0
$58$	6.00000i	0.787839i
$59$	11.0000	1.43208	0.716039	−	0.698060i	$-0.245953\pi$
$59$	11.0000	1.43208	0.716039	+	0.698060i	$0.245953\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	4.00000i	0.508001i
$63$	0	0
$64$	−7.00000	−0.875000
$65$	0	0
$66$	0	0
$67$	2.00000i	0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	3.00000i	0.363803i
$69$	0	0
$70$	0	0
$71$	5.00000	0.593391	0.296695	−	0.954972i	$-0.404115\pi$
$71$	5.00000	0.593391	0.296695	+	0.954972i	$0.404115\pi$
$72$	0	0
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	1.00000	0.114708
$77$	3.00000i	0.341882i
$78$	0	0
$79$	−5.00000	−0.562544	−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000	−0.562544	−0.281272	+	0.959628i	$0.590756\pi$
$80$	0	0
$81$	0	0
$82$	5.00000i	0.552158i
$83$	− 8.00000i	− 0.878114i	−0.898459	−	0.439057i	$-0.855313\pi$
$83$	− 8.00000i	− 0.878114i	0.898459	−	0.439057i	$-0.144687\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	− 3.00000i	− 0.319801i
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	− 1.00000i	− 0.104257i
$93$	0	0
$94$	−3.00000	−0.309426
$95$	0	0
$96$	0	0
$97$	− 17.0000i	− 1.72609i	−0.505128	−	0.863044i	$-0.668555\pi$
$97$	− 17.0000i	− 1.72609i	0.505128	−	0.863044i	$-0.331445\pi$
$98$	− 2.00000i	− 0.202031i
$99$	0	0
$100$	0	0
$101$	11.0000	1.09454	0.547270	−	0.836956i	$-0.315667\pi$
$101$	11.0000	1.09454	0.547270	+	0.836956i	$0.315667\pi$
$102$	0	0
$103$	− 2.00000i	− 0.197066i	−0.995134	−	0.0985329i	$-0.968585\pi$
$103$	− 2.00000i	− 0.197066i	0.995134	−	0.0985329i	$-0.0314150\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	10.0000	0.971286
$107$	18.0000i	1.74013i	0.492941	+	0.870063i	$0.335922\pi$
$107$	18.0000i	1.74013i	−0.492941	+	0.870063i	$0.664078\pi$
$108$	0	0
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	0	0
$111$	0	0
$112$	3.00000i	0.283473i
$113$	− 18.0000i	− 1.69330i	−0.532152	−	0.846649i	$-0.678617\pi$
$113$	− 18.0000i	− 1.69330i	0.532152	−	0.846649i	$-0.321383\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	0	0
$118$	11.0000i	1.01263i
$119$	9.00000	0.825029
$120$	0	0
$121$	1.00000	0.0909091
$122$	14.0000i	1.26750i
$123$	0	0
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	− 5.00000i	− 0.443678i	−0.975083	−	0.221839i	$-0.928794\pi$
$127$	− 5.00000i	− 0.443678i	0.975083	−	0.221839i	$-0.0712060\pi$
$128$	3.00000i	0.265165i
$129$	0	0
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	− 3.00000i	− 0.260133i
$134$	−2.00000	−0.172774
$135$	0	0
$136$	−9.00000	−0.771744
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	5.00000i	0.419591i
$143$	2.00000i	0.167248i
$144$	0	0
$145$	0	0
$146$	2.00000	0.165521
$147$	0	0
$148$	1.00000i	0.0821995i
$149$	7.00000	0.573462	0.286731	−	0.958011i	$-0.407431\pi$
$149$	7.00000	0.573462	0.286731	+	0.958011i	$0.407431\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	3.00000i	0.243332i
$153$	0	0
$154$	−3.00000	−0.241747
$155$	0	0
$156$	0	0
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	− 5.00000i	− 0.397779i
$159$	0	0
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	− 10.0000i	− 0.783260i	−0.920123	−	0.391630i	$-0.871911\pi$
$163$	− 10.0000i	− 0.783260i	0.920123	−	0.391630i	$-0.128089\pi$
$164$	5.00000	0.390434
$165$	0	0
$166$	8.00000	0.620920
$167$	10.0000i	0.773823i	0.922117	+	0.386912i	$0.126458\pi$
$167$	10.0000i	0.773823i	−0.922117	+	0.386912i	$0.873542\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	− 4.00000i	− 0.304997i
$173$	9.00000i	0.684257i	0.939653	+	0.342129i	$0.111148\pi$
$173$	9.00000i	0.684257i	−0.939653	+	0.342129i	$0.888852\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	− 10.0000i	− 0.749532i
$179$	−1.00000	−0.0747435	−0.0373718	−	0.999301i	$-0.511899\pi$
$179$	−1.00000	−0.0747435	−0.0373718	+	0.999301i	$0.511899\pi$
$180$	0	0
$181$	−25.0000	−1.85824	−0.929118	−	0.369784i	$-0.879432\pi$
$181$	−25.0000	−1.85824	−0.929118	+	0.369784i	$0.879432\pi$
$182$	− 6.00000i	− 0.444750i
$183$	0	0
$184$	3.00000	0.221163
$185$	0	0
$186$	0	0
$187$	− 3.00000i	− 0.219382i
$188$	3.00000i	0.218797i
$189$	0	0
$190$	0	0
$191$	−19.0000	−1.37479	−0.687396	−	0.726283i	$-0.741246\pi$
$191$	−19.0000	−1.37479	−0.687396	+	0.726283i	$0.741246\pi$
$192$	0	0
$193$	− 4.00000i	− 0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$	− 4.00000i	− 0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$	17.0000	1.22053
$195$	0	0
$196$	−2.00000	−0.142857
$197$	23.0000i	1.63868i	0.573306	+	0.819341i	$0.305660\pi$
$197$	23.0000i	1.63868i	−0.573306	+	0.819341i	$0.694340\pi$
$198$	0	0
$199$	18.0000	1.27599	0.637993	−	0.770042i	$-0.279765\pi$
$199$	18.0000	1.27599	0.637993	+	0.770042i	$0.279765\pi$
$200$	0	0
$201$	0	0
$202$	11.0000i	0.773957i
$203$	− 18.0000i	− 1.26335i
$204$	0	0
$205$	0	0
$206$	2.00000	0.139347
$207$	0	0
$208$	2.00000i	0.138675i
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	− 10.0000i	− 0.686803i
$213$	0	0
$214$	−18.0000	−1.23045
$215$	0	0
$216$	0	0
$217$	− 12.0000i	− 0.814613i
$218$	12.0000i	0.812743i
$219$	0	0
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	22.0000i	1.47323i	0.676313	+	0.736614i	$0.263577\pi$
$223$	22.0000i	1.47323i	−0.676313	+	0.736614i	$0.736423\pi$
$224$	15.0000	1.00223
$225$	0	0
$226$	18.0000	1.19734
$227$	− 8.00000i	− 0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$	− 8.00000i	− 0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$	0	0
$229$	−11.0000	−0.726900	−0.363450	−	0.931614i	$-0.618401\pi$
$229$	−11.0000	−0.726900	−0.363450	+	0.931614i	$0.618401\pi$
$230$	0	0
$231$	0	0
$232$	18.0000i	1.18176i
$233$	− 9.00000i	− 0.589610i	−0.955557	−	0.294805i	$-0.904745\pi$
$233$	− 9.00000i	− 0.589610i	0.955557	−	0.294805i	$-0.0952546\pi$
$234$	0	0
$235$	0	0
$236$	11.0000	0.716039
$237$	0	0
$238$	9.00000i	0.583383i
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	1.00000i	0.0642824i
$243$	0	0
$244$	14.0000	0.896258
$245$	0	0
$246$	0	0
$247$	− 2.00000i	− 0.127257i
$248$	12.0000i	0.762001i
$249$	0	0
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	1.00000i	0.0628695i
$254$	5.00000	0.313728
$255$	0	0
$256$	−17.0000	−1.06250
$257$	12.0000i	0.748539i	0.927320	+	0.374270i	$0.122107\pi$
$257$	12.0000i	0.748539i	−0.927320	+	0.374270i	$0.877893\pi$
$258$	0	0
$259$	3.00000	0.186411
$260$	0	0
$261$	0	0
$262$	− 12.0000i	− 0.741362i
$263$	18.0000i	1.10993i	0.831875	+	0.554964i	$0.187268\pi$
$263$	18.0000i	1.10993i	−0.831875	+	0.554964i	$0.812732\pi$
$264$	0	0
$265$	0	0
$266$	3.00000	0.183942
$267$	0	0
$268$	2.00000i	0.122169i
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	−3.00000	−0.182237	−0.0911185	−	0.995840i	$-0.529044\pi$
$271$	−3.00000	−0.182237	−0.0911185	+	0.995840i	$0.529044\pi$
$272$	− 3.00000i	− 0.181902i
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.0000	1.25275	0.626377	−	0.779520i	$-0.284537\pi$
$281$	21.0000	1.25275	0.626377	+	0.779520i	$0.284537\pi$
$282$	0	0
$283$	23.0000i	1.36721i	0.729853	+	0.683604i	$0.239588\pi$
$283$	23.0000i	1.36721i	−0.729853	+	0.683604i	$0.760412\pi$
$284$	5.00000	0.296695
$285$	0	0
$286$	−2.00000	−0.118262
$287$	− 15.0000i	− 0.885422i
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	− 2.00000i	− 0.117041i
$293$	− 13.0000i	− 0.759468i	−0.925096	−	0.379734i	$-0.876015\pi$
$293$	− 13.0000i	− 0.759468i	0.925096	−	0.379734i	$-0.123985\pi$
$294$	0	0
$295$	0	0
$296$	−3.00000	−0.174371
$297$	0	0
$298$	7.00000i	0.405499i
$299$	−2.00000	−0.115663
$300$	0	0
$301$	−12.0000	−0.691669
$302$	− 16.0000i	− 0.920697i
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	32.0000i	1.82634i	0.407583	+	0.913168i	$0.366372\pi$
$307$	32.0000i	1.82634i	−0.407583	+	0.913168i	$0.633628\pi$
$308$	3.00000i	0.170941i
$309$	0	0
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	1.00000i	0.0565233i	0.999601	+	0.0282617i	$0.00899717\pi$
$313$	1.00000i	0.0565233i	−0.999601	+	0.0282617i	$0.991003\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−5.00000	−0.281272
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	0	0
$321$	0	0
$322$	− 3.00000i	− 0.167183i
$323$	3.00000i	0.166924i
$324$	0	0
$325$	0	0
$326$	10.0000	0.553849
$327$	0	0
$328$	15.0000i	0.828236i
$329$	9.00000	0.496186
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	− 8.00000i	− 0.439057i
$333$	0	0
$334$	−10.0000	−0.547176
$335$	0	0
$336$	0	0
$337$	6.00000i	0.326841i	0.986557	+	0.163420i	$0.0522527\pi$
$337$	6.00000i	0.326841i	−0.986557	+	0.163420i	$0.947747\pi$
$338$	9.00000i	0.489535i
$339$	0	0
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	− 15.0000i	− 0.809924i
$344$	12.0000	0.646997
$345$	0	0
$346$	−9.00000	−0.483843
$347$	− 10.0000i	− 0.536828i	−0.963304	−	0.268414i	$-0.913500\pi$
$347$	− 10.0000i	− 0.536828i	0.963304	−	0.268414i	$-0.0864995\pi$
$348$	0	0
$349$	−8.00000	−0.428230	−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000	−0.428230	−0.214115	+	0.976808i	$0.568687\pi$
$350$	0	0
$351$	0	0
$352$	− 5.00000i	− 0.266501i
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	0	0
$355$	0	0
$356$	−10.0000	−0.529999
$357$	0	0
$358$	− 1.00000i	− 0.0528516i
$359$	−28.0000	−1.47778	−0.738892	−	0.673824i	$-0.764651\pi$
$359$	−28.0000	−1.47778	−0.738892	+	0.673824i	$0.764651\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	− 25.0000i	− 1.31397i
$363$	0	0
$364$	−6.00000	−0.314485
$365$	0	0
$366$	0	0
$367$	4.00000i	0.208798i	0.994535	+	0.104399i	$0.0332919\pi$
$367$	4.00000i	0.208798i	−0.994535	+	0.104399i	$0.966708\pi$
$368$	1.00000i	0.0521286i
$369$	0	0
$370$	0	0
$371$	−30.0000	−1.55752
$372$	0	0
$373$	− 6.00000i	− 0.310668i	−0.987862	−	0.155334i	$-0.950355\pi$
$373$	− 6.00000i	− 0.310668i	0.987862	−	0.155334i	$-0.0496454\pi$
$374$	3.00000	0.155126
$375$	0	0
$376$	−9.00000	−0.464140
$377$	− 12.0000i	− 0.618031i
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	0	0
$382$	− 19.0000i	− 0.972125i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	4.00000	0.203595
$387$	0	0
$388$	− 17.0000i	− 0.863044i
$389$	−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
$391$	3.00000	0.151717
$392$	− 6.00000i	− 0.303046i
$393$	0	0
$394$	−23.0000	−1.15872
$395$	0	0
$396$	0	0
$397$	− 6.00000i	− 0.301131i	−0.988600	−	0.150566i	$-0.951890\pi$
$397$	− 6.00000i	− 0.301131i	0.988600	−	0.150566i	$-0.0481095\pi$
$398$	18.0000i	0.902258i
$399$	0	0
$400$	0	0
$401$	38.0000	1.89763	0.948815	−	0.315833i	$-0.102284\pi$
$401$	38.0000	1.89763	0.948815	+	0.315833i	$0.102284\pi$
$402$	0	0
$403$	− 8.00000i	− 0.398508i
$404$	11.0000	0.547270
$405$	0	0
$406$	18.0000	0.893325
$407$	− 1.00000i	− 0.0495682i
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	0	0
$412$	− 2.00000i	− 0.0985329i
$413$	− 33.0000i	− 1.62382i
$414$	0	0
$415$	0	0
$416$	10.0000	0.490290
$417$	0	0
$418$	− 1.00000i	− 0.0489116i
$419$	−23.0000	−1.12362	−0.561812	−	0.827265i	$-0.689895\pi$
$419$	−23.0000	−1.12362	−0.561812	+	0.827265i	$0.689895\pi$
$420$	0	0
$421$	9.00000	0.438633	0.219317	−	0.975654i	$-0.429617\pi$
$421$	9.00000	0.438633	0.219317	+	0.975654i	$0.429617\pi$
$422$	4.00000i	0.194717i
$423$	0	0
$424$	30.0000	1.45693
$425$	0	0
$426$	0	0
$427$	− 42.0000i	− 2.03252i
$428$	18.0000i	0.870063i
$429$	0	0
$430$	0	0
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	0	0
$433$	− 2.00000i	− 0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	12.0000	0.576018
$435$	0	0
$436$	12.0000	0.574696
$437$	− 1.00000i	− 0.0478365i
$438$	0	0
$439$	35.0000	1.67046	0.835229	−	0.549902i	$-0.185335\pi$
$439$	35.0000	1.67046	0.835229	+	0.549902i	$0.185335\pi$
$440$	0	0
$441$	0	0
$442$	6.00000i	0.285391i
$443$	31.0000i	1.47285i	0.676517	+	0.736427i	$0.263489\pi$
$443$	31.0000i	1.47285i	−0.676517	+	0.736427i	$0.736511\pi$
$444$	0	0
$445$	0	0
$446$	−22.0000	−1.04173
$447$	0	0
$448$	21.0000i	0.992157i
$449$	−8.00000	−0.377543	−0.188772	−	0.982021i	$-0.560451\pi$
$449$	−8.00000	−0.377543	−0.188772	+	0.982021i	$0.560451\pi$
$450$	0	0
$451$	−5.00000	−0.235441
$452$	− 18.0000i	− 0.846649i
$453$	0	0
$454$	8.00000	0.375459
$455$	0	0
$456$	0	0
$457$	28.0000i	1.30978i	0.755722	+	0.654892i	$0.227286\pi$
$457$	28.0000i	1.30978i	−0.755722	+	0.654892i	$0.772714\pi$
$458$	− 11.0000i	− 0.513996i
$459$	0	0
$460$	0	0
$461$	−34.0000	−1.58354	−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000	−1.58354	−0.791769	+	0.610821i	$0.790840\pi$
$462$	0	0
$463$	− 24.0000i	− 1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$	− 24.0000i	− 1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	9.00000	0.416917
$467$	− 36.0000i	− 1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$	− 36.0000i	− 1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$	0	0
$469$	6.00000	0.277054
$470$	0	0
$471$	0	0
$472$	33.0000i	1.51895i
$473$	4.00000i	0.183920i
$474$	0	0
$475$	0	0
$476$	9.00000	0.412514
$477$	0	0
$478$	− 20.0000i	− 0.914779i
$479$	−18.0000	−0.822441	−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000	−0.822441	−0.411220	+	0.911536i	$0.634897\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	4.00000i	0.182195i
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	0	0
$487$	− 26.0000i	− 1.17817i	−0.808070	−	0.589086i	$-0.799488\pi$
$487$	− 26.0000i	− 1.17817i	0.808070	−	0.589086i	$-0.200512\pi$
$488$	42.0000i	1.90125i
$489$	0	0
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	18.0000i	0.810679i
$494$	2.00000	0.0899843
$495$	0	0
$496$	−4.00000	−0.179605
$497$	− 15.0000i	− 0.672842i
$498$	0	0
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	0	0
$501$	0	0
$502$	− 12.0000i	− 0.535586i
$503$	− 20.0000i	− 0.891756i	−0.895094	−	0.445878i	$-0.852892\pi$
$503$	− 20.0000i	− 0.891756i	0.895094	−	0.445878i	$-0.147108\pi$
$504$	0	0
$505$	0	0
$506$	−1.00000	−0.0444554
$507$	0	0
$508$	− 5.00000i	− 0.221839i
$509$	−26.0000	−1.15243	−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000	−1.15243	−0.576215	+	0.817298i	$0.695471\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	− 11.0000i	− 0.486136i
$513$	0	0
$514$	−12.0000	−0.529297
$515$	0	0
$516$	0	0
$517$	− 3.00000i	− 0.131940i
$518$	3.00000i	0.131812i
$519$	0	0
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	0	0
$523$	− 7.00000i	− 0.306089i	−0.988219	−	0.153044i	$-0.951092\pi$
$523$	− 7.00000i	− 0.306089i	0.988219	−	0.153044i	$-0.0489077\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−18.0000	−0.784837
$527$	12.0000i	0.522728i
$528$	0	0
$529$	22.0000	0.956522
$530$	0	0
$531$	0	0
$532$	− 3.00000i	− 0.130066i
$533$	− 10.0000i	− 0.433148i
$534$	0	0
$535$	0	0
$536$	−6.00000	−0.259161
$537$	0	0
$538$	− 24.0000i	− 1.03471i
$539$	2.00000	0.0861461
$540$	0	0
$541$	32.0000	1.37579	0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000	1.37579	0.687894	+	0.725811i	$0.258536\pi$
$542$	− 3.00000i	− 0.128861i
$543$	0	0
$544$	−15.0000	−0.643120
$545$	0	0
$546$	0	0
$547$	− 33.0000i	− 1.41098i	−0.708721	−	0.705489i	$-0.750727\pi$
$547$	− 33.0000i	− 1.41098i	0.708721	−	0.705489i	$-0.249273\pi$
$548$	6.00000i	0.256307i
$549$	0	0
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	15.0000i	0.637865i
$554$	−22.0000	−0.934690
$555$	0	0
$556$	0	0
$557$	22.0000i	0.932170i	0.884740	+	0.466085i	$0.154336\pi$
$557$	22.0000i	0.932170i	−0.884740	+	0.466085i	$0.845664\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	21.0000i	0.885832i
$563$	− 18.0000i	− 0.758610i	−0.925272	−	0.379305i	$-0.876163\pi$
$563$	− 18.0000i	− 0.758610i	0.925272	−	0.379305i	$-0.123837\pi$
$564$	0	0
$565$	0	0
$566$	−23.0000	−0.966762
$567$	0	0
$568$	15.0000i	0.629386i
$569$	37.0000	1.55112	0.775560	−	0.631273i	$-0.217467\pi$
$569$	37.0000	1.55112	0.775560	+	0.631273i	$0.217467\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	2.00000i	0.0836242i
$573$	0	0
$574$	15.0000	0.626088
$575$	0	0
$576$	0	0
$577$	− 43.0000i	− 1.79011i	−0.445952	−	0.895057i	$-0.647135\pi$
$577$	− 43.0000i	− 1.79011i	0.445952	−	0.895057i	$-0.352865\pi$
$578$	8.00000i	0.332756i
$579$	0	0
$580$	0	0
$581$	−24.0000	−0.995688
$582$	0	0
$583$	10.0000i	0.414158i
$584$	6.00000	0.248282
$585$	0	0
$586$	13.0000	0.537025
$587$	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
$589$	4.00000	0.164817
$590$	0	0
$591$	0	0
$592$	− 1.00000i	− 0.0410997i
$593$	14.0000i	0.574911i	0.957794	+	0.287456i	$0.0928094\pi$
$593$	14.0000i	0.574911i	−0.957794	+	0.287456i	$0.907191\pi$
$594$	0	0
$595$	0	0
$596$	7.00000	0.286731
$597$	0	0
$598$	− 2.00000i	− 0.0817861i
$599$	−27.0000	−1.10319	−0.551595	−	0.834112i	$-0.685981\pi$
$599$	−27.0000	−1.10319	−0.551595	+	0.834112i	$0.685981\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	− 12.0000i	− 0.489083i
$603$	0	0
$604$	−16.0000	−0.651031
$605$	0	0
$606$	0	0
$607$	40.0000i	1.62355i	0.583970	+	0.811775i	$0.301498\pi$
$607$	40.0000i	1.62355i	−0.583970	+	0.811775i	$0.698502\pi$
$608$	5.00000i	0.202777i
$609$	0	0
$610$	0	0
$611$	6.00000	0.242734
$612$	0	0
$613$	− 16.0000i	− 0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$	− 16.0000i	− 0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$	−32.0000	−1.29141
$615$	0	0
$616$	−9.00000	−0.362620
$617$	− 28.0000i	− 1.12724i	−0.826035	−	0.563619i	$-0.809409\pi$
$617$	− 28.0000i	− 1.12724i	0.826035	−	0.563619i	$-0.190591\pi$
$618$	0	0
$619$	−34.0000	−1.36658	−0.683288	−	0.730149i	$-0.739451\pi$
$619$	−34.0000	−1.36658	−0.683288	+	0.730149i	$0.739451\pi$
$620$	0	0
$621$	0	0
$622$	12.0000i	0.481156i
$623$	30.0000i	1.20192i
$624$	0	0
$625$	0	0
$626$	−1.00000	−0.0399680
$627$	0	0
$628$	10.0000i	0.399043i
$629$	−3.00000	−0.119618
$630$	0	0
$631$	34.0000	1.35352	0.676759	−	0.736204i	$-0.263384\pi$
$631$	34.0000	1.35352	0.676759	+	0.736204i	$0.263384\pi$
$632$	− 15.0000i	− 0.596668i
$633$	0	0
$634$	6.00000	0.238290
$635$	0	0
$636$	0	0
$637$	4.00000i	0.158486i
$638$	− 6.00000i	− 0.237542i
$639$	0	0
$640$	0	0
$641$	16.0000	0.631962	0.315981	−	0.948766i	$-0.397666\pi$
$641$	16.0000	0.631962	0.315981	+	0.948766i	$0.397666\pi$
$642$	0	0
$643$	− 14.0000i	− 0.552106i	−0.961142	−	0.276053i	$-0.910973\pi$
$643$	− 14.0000i	− 0.552106i	0.961142	−	0.276053i	$-0.0890266\pi$
$644$	−3.00000	−0.118217
$645$	0	0
$646$	−3.00000	−0.118033
$647$	− 13.0000i	− 0.511083i	−0.966798	−	0.255541i	$-0.917746\pi$
$647$	− 13.0000i	− 0.511083i	0.966798	−	0.255541i	$-0.0822537\pi$
$648$	0	0
$649$	−11.0000	−0.431788
$650$	0	0
$651$	0	0
$652$	− 10.0000i	− 0.391630i
$653$	4.00000i	0.156532i	0.996933	+	0.0782660i	$0.0249384\pi$
$653$	4.00000i	0.156532i	−0.996933	+	0.0782660i	$0.975062\pi$
$654$	0	0
$655$	0	0
$656$	−5.00000	−0.195217
$657$	0	0
$658$	9.00000i	0.350857i
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	0	0
$661$	35.0000	1.36134	0.680671	−	0.732589i	$-0.261688\pi$
$661$	35.0000	1.36134	0.680671	+	0.732589i	$0.261688\pi$
$662$	4.00000i	0.155464i
$663$	0	0
$664$	24.0000	0.931381
$665$	0	0
$666$	0	0
$667$	− 6.00000i	− 0.232321i
$668$	10.0000i	0.386912i
$669$	0	0
$670$	0	0
$671$	−14.0000	−0.540464
$672$	0	0
$673$	− 4.00000i	− 0.154189i	−0.997024	−	0.0770943i	$-0.975436\pi$
$673$	− 4.00000i	− 0.154189i	0.997024	−	0.0770943i	$-0.0245643\pi$
$674$	−6.00000	−0.231111
$675$	0	0
$676$	9.00000	0.346154
$677$	− 14.0000i	− 0.538064i	−0.963131	−	0.269032i	$-0.913296\pi$
$677$	− 14.0000i	− 0.538064i	0.963131	−	0.269032i	$-0.0867037\pi$
$678$	0	0
$679$	−51.0000	−1.95720
$680$	0	0
$681$	0	0
$682$	− 4.00000i	− 0.153168i
$683$	− 47.0000i	− 1.79841i	−0.437533	−	0.899203i	$-0.644148\pi$
$683$	− 47.0000i	− 1.79841i	0.437533	−	0.899203i	$-0.355852\pi$
$684$	0	0
$685$	0	0
$686$	15.0000	0.572703
$687$	0	0
$688$	4.00000i	0.152499i
$689$	−20.0000	−0.761939
$690$	0	0
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	9.00000i	0.342129i
$693$	0	0
$694$	10.0000	0.379595
$695$	0	0
$696$	0	0
$697$	15.0000i	0.568166i
$698$	− 8.00000i	− 0.302804i
$699$	0	0
$700$	0	0
$701$	−23.0000	−0.868698	−0.434349	−	0.900745i	$-0.643022\pi$
$701$	−23.0000	−0.868698	−0.434349	+	0.900745i	$0.643022\pi$
$702$	0	0
$703$	1.00000i	0.0377157i
$704$	7.00000	0.263822
$705$	0	0
$706$	6.00000	0.225813
$707$	− 33.0000i	− 1.24109i
$708$	0	0
$709$	−35.0000	−1.31445	−0.657226	−	0.753693i	$-0.728270\pi$
$709$	−35.0000	−1.31445	−0.657226	+	0.753693i	$0.728270\pi$
$710$	0	0
$711$	0	0
$712$	− 30.0000i	− 1.12430i
$713$	− 4.00000i	− 0.149801i
$714$	0	0
$715$	0	0
$716$	−1.00000	−0.0373718
$717$	0	0
$718$	− 28.0000i	− 1.04495i
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	−6.00000	−0.223452
$722$	− 18.0000i	− 0.669891i
$723$	0	0
$724$	−25.0000	−0.929118
$725$	0	0
$726$	0	0
$727$	28.0000i	1.03846i	0.854634	+	0.519231i	$0.173782\pi$
$727$	28.0000i	1.03846i	−0.854634	+	0.519231i	$0.826218\pi$
$728$	− 18.0000i	− 0.667124i
$729$	0	0
$730$	0	0
$731$	12.0000	0.443836
$732$	0	0
$733$	6.00000i	0.221615i	0.993842	+	0.110808i	$0.0353437\pi$
$733$	6.00000i	0.221615i	−0.993842	+	0.110808i	$0.964656\pi$
$734$	−4.00000	−0.147643
$735$	0	0
$736$	5.00000	0.184302
$737$	− 2.00000i	− 0.0736709i
$738$	0	0
$739$	−1.00000	−0.0367856	−0.0183928	−	0.999831i	$-0.505855\pi$
$739$	−1.00000	−0.0367856	−0.0183928	+	0.999831i	$0.505855\pi$
$740$	0	0
$741$	0	0
$742$	− 30.0000i	− 1.10133i
$743$	14.0000i	0.513610i	0.966463	+	0.256805i	$0.0826698\pi$
$743$	14.0000i	0.513610i	−0.966463	+	0.256805i	$0.917330\pi$
$744$	0	0
$745$	0	0
$746$	6.00000	0.219676
$747$	0	0
$748$	− 3.00000i	− 0.109691i
$749$	54.0000	1.97312
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	− 3.00000i	− 0.109399i
$753$	0	0
$754$	12.0000	0.437014
$755$	0	0
$756$	0	0
$757$	− 42.0000i	− 1.52652i	−0.646094	−	0.763258i	$-0.723599\pi$
$757$	− 42.0000i	− 1.52652i	0.646094	−	0.763258i	$-0.276401\pi$
$758$	16.0000i	0.581146i
$759$	0	0
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	− 36.0000i	− 1.30329i
$764$	−19.0000	−0.687396
$765$	0	0
$766$	0	0
$767$	− 22.0000i	− 0.794374i
$768$	0	0
$769$	28.0000	1.00971	0.504853	−	0.863205i	$-0.331547\pi$
$769$	28.0000	1.00971	0.504853	+	0.863205i	$0.331547\pi$
$770$	0	0
$771$	0	0
$772$	− 4.00000i	− 0.143963i
$773$	− 44.0000i	− 1.58257i	−0.611448	−	0.791285i	$-0.709412\pi$
$773$	− 44.0000i	− 1.58257i	0.611448	−	0.791285i	$-0.290588\pi$
$774$	0	0
$775$	0	0
$776$	51.0000	1.83079
$777$	0	0
$778$	− 6.00000i	− 0.215110i
$779$	5.00000	0.179144
$780$	0	0
$781$	−5.00000	−0.178914
$782$	3.00000i	0.107280i
$783$	0	0
$784$	2.00000	0.0714286
$785$	0	0
$786$	0	0
$787$	43.0000i	1.53278i	0.642373	+	0.766392i	$0.277950\pi$
$787$	43.0000i	1.53278i	−0.642373	+	0.766392i	$0.722050\pi$
$788$	23.0000i	0.819341i
$789$	0	0
$790$	0	0
$791$	−54.0000	−1.92002
$792$	0	0
$793$	− 28.0000i	− 0.994309i
$794$	6.00000	0.212932
$795$	0	0
$796$	18.0000	0.637993
$797$	22.0000i	0.779280i	0.920967	+	0.389640i	$0.127401\pi$
$797$	22.0000i	0.779280i	−0.920967	+	0.389640i	$0.872599\pi$
$798$	0	0
$799$	−9.00000	−0.318397
$800$	0	0
$801$	0	0
$802$	38.0000i	1.34183i
$803$	2.00000i	0.0705785i
$804$	0	0
$805$	0	0
$806$	8.00000	0.281788
$807$	0	0
$808$	33.0000i	1.16094i
$809$	−15.0000	−0.527372	−0.263686	−	0.964609i	$-0.584938\pi$
$809$	−15.0000	−0.527372	−0.263686	+	0.964609i	$0.584938\pi$
$810$	0	0
$811$	−21.0000	−0.737410	−0.368705	−	0.929547i	$-0.620199\pi$
$811$	−21.0000	−0.737410	−0.368705	+	0.929547i	$0.620199\pi$
$812$	− 18.0000i	− 0.631676i
$813$	0	0
$814$	1.00000	0.0350500
$815$	0	0
$816$	0	0
$817$	− 4.00000i	− 0.139942i
$818$	− 6.00000i	− 0.209785i
$819$	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
$823$	42.0000i	1.46403i	0.681290	+	0.732014i	$0.261419\pi$
$823$	42.0000i	1.46403i	−0.681290	+	0.732014i	$0.738581\pi$
$824$	6.00000	0.209020
$825$	0	0
$826$	33.0000	1.14822
$827$	− 48.0000i	− 1.66912i	−0.550914	−	0.834562i	$-0.685721\pi$
$827$	− 48.0000i	− 1.66912i	0.550914	−	0.834562i	$-0.314279\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	0	0
$832$	14.0000i	0.485363i
$833$	− 6.00000i	− 0.207888i
$834$	0	0
$835$	0	0
$836$	−1.00000	−0.0345857
$837$	0	0
$838$	− 23.0000i	− 0.794522i
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	9.00000i	0.310160i
$843$	0	0
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	− 3.00000i	− 0.103081i
$848$	10.0000i	0.343401i
$849$	0	0
$850$	0	0
$851$	1.00000	0.0342796
$852$	0	0
$853$	40.0000i	1.36957i	0.728743	+	0.684787i	$0.240105\pi$
$853$	40.0000i	1.36957i	−0.728743	+	0.684787i	$0.759895\pi$
$854$	42.0000	1.43721
$855$	0	0
$856$	−54.0000	−1.84568
$857$	39.0000i	1.33221i	0.745856	+	0.666107i	$0.232041\pi$
$857$	39.0000i	1.33221i	−0.745856	+	0.666107i	$0.767959\pi$
$858$	0	0
$859$	26.0000	0.887109	0.443554	−	0.896248i	$-0.353717\pi$
$859$	26.0000	0.887109	0.443554	+	0.896248i	$0.353717\pi$
$860$	0	0
$861$	0	0
$862$	− 36.0000i	− 1.22616i
$863$	− 4.00000i	− 0.136162i	−0.997680	−	0.0680808i	$-0.978312\pi$
$863$	− 4.00000i	− 0.136162i	0.997680	−	0.0680808i	$-0.0216876\pi$
$864$	0	0
$865$	0	0
$866$	2.00000	0.0679628
$867$	0	0
$868$	− 12.0000i	− 0.407307i
$869$	5.00000	0.169613
$870$	0	0
$871$	4.00000	0.135535
$872$	36.0000i	1.21911i
$873$	0	0
$874$	1.00000	0.0338255
$875$	0	0
$876$	0	0
$877$	− 52.0000i	− 1.75592i	−0.478738	−	0.877958i	$-0.658906\pi$
$877$	− 52.0000i	− 1.75592i	0.478738	−	0.877958i	$-0.341094\pi$
$878$	35.0000i	1.18119i
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	− 2.00000i	− 0.0673054i	−0.999434	−	0.0336527i	$-0.989286\pi$
$883$	− 2.00000i	− 0.0673054i	0.999434	−	0.0336527i	$-0.0107140\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	−31.0000	−1.04147
$887$	14.0000i	0.470074i	0.971986	+	0.235037i	$0.0755211\pi$
$887$	14.0000i	0.470074i	−0.971986	+	0.235037i	$0.924479\pi$
$888$	0	0
$889$	−15.0000	−0.503084
$890$	0	0
$891$	0	0
$892$	22.0000i	0.736614i
$893$	3.00000i	0.100391i
$894$	0	0
$895$	0	0
$896$	9.00000	0.300669
$897$	0	0
$898$	− 8.00000i	− 0.266963i
$899$	24.0000	0.800445
$900$	0	0
$901$	30.0000	0.999445
$902$	− 5.00000i	− 0.166482i
$903$	0	0
$904$	54.0000	1.79601
$905$	0	0
$906$	0	0
$907$	2.00000i	0.0664089i	0.999449	+	0.0332045i	$0.0105712\pi$
$907$	2.00000i	0.0664089i	−0.999449	+	0.0332045i	$0.989429\pi$
$908$	− 8.00000i	− 0.265489i
$909$	0	0
$910$	0	0
$911$	53.0000	1.75597	0.877984	−	0.478690i	$-0.158888\pi$
$911$	53.0000	1.75597	0.877984	+	0.478690i	$0.158888\pi$
$912$	0	0
$913$	8.00000i	0.264761i
$914$	−28.0000	−0.926158
$915$	0	0
$916$	−11.0000	−0.363450
$917$	36.0000i	1.18882i
$918$	0	0
$919$	39.0000	1.28649	0.643246	−	0.765660i	$-0.277587\pi$
$919$	39.0000	1.28649	0.643246	+	0.765660i	$0.277587\pi$
$920$	0	0
$921$	0	0
$922$	− 34.0000i	− 1.11973i
$923$	− 10.0000i	− 0.329154i
$924$	0	0
$925$	0	0
$926$	24.0000	0.788689
$927$	0	0
$928$	30.0000i	0.984798i
$929$	32.0000	1.04989	0.524943	−	0.851137i	$-0.324087\pi$
$929$	32.0000	1.04989	0.524943	+	0.851137i	$0.324087\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	− 9.00000i	− 0.294805i
$933$	0	0
$934$	36.0000	1.17796
$935$	0	0
$936$	0	0
$937$	18.0000i	0.588034i	0.955800	+	0.294017i	$0.0949923\pi$
$937$	18.0000i	0.588034i	−0.955800	+	0.294017i	$0.905008\pi$
$938$	6.00000i	0.195907i
$939$	0	0
$940$	0	0
$941$	−5.00000	−0.162995	−0.0814977	−	0.996674i	$-0.525970\pi$
$941$	−5.00000	−0.162995	−0.0814977	+	0.996674i	$0.525970\pi$
$942$	0	0
$943$	− 5.00000i	− 0.162822i
$944$	−11.0000	−0.358020
$945$	0	0
$946$	−4.00000	−0.130051
$947$	− 27.0000i	− 0.877382i	−0.898638	−	0.438691i	$-0.855442\pi$
$947$	− 27.0000i	− 0.877382i	0.898638	−	0.438691i	$-0.144558\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	0	0
$952$	27.0000i	0.875075i
$953$	21.0000i	0.680257i	0.940379	+	0.340128i	$0.110471\pi$
$953$	21.0000i	0.680257i	−0.940379	+	0.340128i	$0.889529\pi$
$954$	0	0
$955$	0	0
$956$	−20.0000	−0.646846
$957$	0	0
$958$	− 18.0000i	− 0.581554i
$959$	18.0000	0.581250
$960$	0	0
$961$	−15.0000	−0.483871
$962$	2.00000i	0.0644826i
$963$	0	0
$964$	4.00000	0.128831
$965$	0	0
$966$	0	0
$967$	20.0000i	0.643157i	0.946883	+	0.321578i	$0.104213\pi$
$967$	20.0000i	0.643157i	−0.946883	+	0.321578i	$0.895787\pi$
$968$	3.00000i	0.0964237i
$969$	0	0
$970$	0	0
$971$	−21.0000	−0.673922	−0.336961	−	0.941519i	$-0.609399\pi$
$971$	−21.0000	−0.673922	−0.336961	+	0.941519i	$0.609399\pi$
$972$	0	0
$973$	0	0
$974$	26.0000	0.833094
$975$	0	0
$976$	−14.0000	−0.448129
$977$	46.0000i	1.47167i	0.677161	+	0.735835i	$0.263210\pi$
$977$	46.0000i	1.47167i	−0.677161	+	0.735835i	$0.736790\pi$
$978$	0	0
$979$	10.0000	0.319601
$980$	0	0
$981$	0	0
$982$	− 28.0000i	− 0.893516i
$983$	49.0000i	1.56286i	0.623995	+	0.781429i	$0.285509\pi$
$983$	49.0000i	1.56286i	−0.623995	+	0.781429i	$0.714491\pi$
$984$	0	0
$985$	0	0
$986$	−18.0000	−0.573237
$987$	0	0
$988$	− 2.00000i	− 0.0636285i
$989$	−4.00000	−0.127193
$990$	0	0
$991$	−34.0000	−1.08005	−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000	−1.08005	−0.540023	+	0.841650i	$0.681584\pi$
$992$	20.0000i	0.635001i
$993$	0	0
$994$	15.0000	0.475771
$995$	0	0
$996$	0	0
$997$	50.0000i	1.58352i	0.610835	+	0.791758i	$0.290834\pi$
$997$	50.0000i	1.58352i	−0.610835	+	0.791758i	$0.709166\pi$
$998$	− 24.0000i	− 0.759707i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.c.c.199.2		2
3.2	odd	2		2475.2.c.e.199.1		2
5.2	odd	4		2475.2.a.d.1.1	yes	1
5.3	odd	4		2475.2.a.h.1.1	yes	1
5.4	even	2	inner	2475.2.c.c.199.1		2
15.2	even	4		2475.2.a.k.1.1	yes	1
15.8	even	4		2475.2.a.b.1.1	✓	1
15.14	odd	2		2475.2.c.e.199.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2475.2.a.b.1.1	✓	1	15.8	even	4
2475.2.a.d.1.1	yes	1	5.2	odd	4
2475.2.a.h.1.1	yes	1	5.3	odd	4
2475.2.a.k.1.1	yes	1	15.2	even	4
2475.2.c.c.199.1		2	5.4	even	2	inner
2475.2.c.c.199.2		2	1.1	even	1	trivial
2475.2.c.e.199.1		2	3.2	odd	2
2475.2.c.e.199.2		2	15.14	odd	2

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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{4} -3.00000i q^{7} +3.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{4} -3.00000i q^{7} +3.00000i q^{8} -1.00000 q^{11} -2.00000i q^{13} +3.00000 q^{14} -1.00000 q^{16} +3.00000i q^{17} +1.00000 q^{19} -1.00000i q^{22} -1.00000i q^{23} +2.00000 q^{26} -3.00000i q^{28} +6.00000 q^{29} +4.00000 q^{31} +5.00000i q^{32} -3.00000 q^{34} +1.00000i q^{37} +1.00000i q^{38} +5.00000 q^{41} -4.00000i q^{43} -1.00000 q^{44} +1.00000 q^{46} +3.00000i q^{47} -2.00000 q^{49} -2.00000i q^{52} -10.0000i q^{53} +9.00000 q^{56} +6.00000i q^{58} +11.0000 q^{59} +14.0000 q^{61} +4.00000i q^{62} -7.00000 q^{64} +2.00000i q^{67} +3.00000i q^{68} +5.00000 q^{71} -2.00000i q^{73} -1.00000 q^{74} +1.00000 q^{76} +3.00000i q^{77} -5.00000 q^{79} +5.00000i q^{82} -8.00000i q^{83} +4.00000 q^{86} -3.00000i q^{88} -10.0000 q^{89} -6.00000 q^{91} -1.00000i q^{92} -3.00000 q^{94} -17.0000i q^{97} -2.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4}+O(q^{10}) \) 2 * q + 2 * q^4	\( 2 q + 2 q^{4} - 2 q^{11} + 6 q^{14} - 2 q^{16} + 2 q^{19} + 4 q^{26} + 12 q^{29} + 8 q^{31} - 6 q^{34} + 10 q^{41} - 2 q^{44} + 2 q^{46} - 4 q^{49} + 18 q^{56} + 22 q^{59} + 28 q^{61} - 14 q^{64} + 10 q^{71} - 2 q^{74} + 2 q^{76} - 10 q^{79} + 8 q^{86} - 20 q^{89} - 12 q^{91} - 6 q^{94}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^11 + 6 * q^14 - 2 * q^16 + 2 * q^19 + 4 * q^26 + 12 * q^29 + 8 * q^31 - 6 * q^34 + 10 * q^41 - 2 * q^44 + 2 * q^46 - 4 * q^49 + 18 * q^56 + 22 * q^59 + 28 * q^61 - 14 * q^64 + 10 * q^71 - 2 * q^74 + 2 * q^76 - 10 * q^79 + 8 * q^86 - 20 * q^89 - 12 * q^91 - 6 * q^94

Introduction

L-functions

Modular forms

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