LMFDB - Embedded newform 7500.2.d.c.1249.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7500,2,Mod(1249,7500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7500, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7500.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7500 = 2^{2} \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7500.d (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:	$59.8878015160$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.6724000000.12
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 16x^{6} + 86x^{4} + 181x^{2} + 121 $ x^8 + 16x^6 + 86x^4 + 181*x^2 + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 300)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.5
Root			$-1.12233i$ of defining polynomial
Character	$\chi$	$=$	7500.1249
Dual form			7500.2.d.c.1249.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.74037i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -1.74037i q^{7} -1.00000 q^{9} +2.32027 q^{11} -4.93831i q^{13} +3.74037i q^{17} +1.69364 q^{19} +1.74037 q^{21} -8.67867i q^{23} -1.00000i q^{27} -3.12233 q^{29} +9.56166 q^{31} +2.32027i q^{33} +5.36700i q^{37} +4.93831 q^{39} -8.72540 q^{41} -2.86270i q^{43} -8.46248i q^{47} +3.97112 q^{49} -3.74037 q^{51} +1.34449i q^{53} +1.69364i q^{57} -4.38793 q^{59} -15.3316 q^{61} +1.74037i q^{63} +9.23075i q^{67} +8.67867 q^{69} +0.0235645 q^{71} +1.02251i q^{73} -4.03813i q^{77} +0.798776 q^{79} +1.00000 q^{81} -2.37008i q^{83} -3.12233i q^{87} +4.86205 q^{89} -8.59447 q^{91} +9.56166i q^{93} -3.46151i q^{97} -2.32027 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} - 2 q^{11} + 10 q^{19} - 8 q^{21} - 12 q^{29} + 22 q^{31} + 10 q^{39} + 8 q^{49} - 8 q^{51} - 2 q^{59} - 44 q^{61} + 18 q^{69} + 40 q^{71} + 6 q^{79} + 8 q^{81} + 30 q^{89} - 20 q^{91} + 2 q^{99}+O(q^{100}) $ 8 * q - 8 * q^9 - 2 * q^11 + 10 * q^19 - 8 * q^21 - 12 * q^29 + 22 * q^31 + 10 * q^39 + 8 * q^49 - 8 * q^51 - 2 * q^59 - 44 * q^61 + 18 * q^69 + 40 * q^71 + 6 * q^79 + 8 * q^81 + 30 * q^89 - 20 * q^91 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7500\mathbb{Z}\right)^\times$.

$n$	$2501$	$3751$	$6877$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.74037i	− 0.657797i	−0.944365	−	0.328899i	$-0.893323\pi$
$7$	− 1.74037i	− 0.657797i	0.944365	−	0.328899i	$-0.106677\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.32027	0.699589	0.349794	−	0.936827i	$-0.386251\pi$
$11$	2.32027	0.699589	0.349794	+	0.936827i	$0.386251\pi$
$12$	0	0
$13$	− 4.93831i	− 1.36964i	−0.728712	−	0.684820i	$-0.759881\pi$
$13$	− 4.93831i	− 1.36964i	0.728712	−	0.684820i	$-0.240119\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.74037i	0.907172i	0.891212	+	0.453586i	$0.149856\pi$
$17$	3.74037i	0.907172i	−0.891212	+	0.453586i	$0.850144\pi$
$18$	0	0
$19$	1.69364	0.388548	0.194274	−	0.980947i	$-0.437765\pi$
$19$	1.69364	0.388548	0.194274	+	0.980947i	$0.437765\pi$
$20$	0	0
$21$	1.74037	0.379779
$22$	0	0
$23$	− 8.67867i	− 1.80963i	−0.425806	−	0.904814i	$-0.640009\pi$
$23$	− 8.67867i	− 1.80963i	0.425806	−	0.904814i	$-0.359991\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−3.12233	−0.579803	−0.289901	−	0.957057i	$-0.593623\pi$
$29$	−3.12233	−0.579803	−0.289901	+	0.957057i	$0.593623\pi$
$30$	0	0
$31$	9.56166	1.71732	0.858662	−	0.512542i	$-0.171296\pi$
$31$	9.56166	1.71732	0.858662	+	0.512542i	$0.171296\pi$
$32$	0	0
$33$	2.32027i	0.403908i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.36700i	0.882329i	0.897426	+	0.441165i	$0.145435\pi$
$37$	5.36700i	0.882329i	−0.897426	+	0.441165i	$0.854565\pi$
$38$	0	0
$39$	4.93831	0.790762
$40$	0	0
$41$	−8.72540	−1.36268	−0.681339	−	0.731968i	$-0.738602\pi$
$41$	−8.72540	−1.36268	−0.681339	+	0.731968i	$0.738602\pi$
$42$	0	0
$43$	− 2.86270i	− 0.436558i	−0.975886	−	0.218279i	$-0.929956\pi$
$43$	− 2.86270i	− 0.436558i	0.975886	−	0.218279i	$-0.0700442\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8.46248i	− 1.23438i	−0.786814	−	0.617190i	$-0.788271\pi$
$47$	− 8.46248i	− 1.23438i	0.786814	−	0.617190i	$-0.211729\pi$
$48$	0	0
$49$	3.97112	0.567303
$50$	0	0
$51$	−3.74037	−0.523756
$52$	0	0
$53$	1.34449i	0.184680i	0.995728	+	0.0923398i	$0.0294346\pi$
$53$	1.34449i	0.184680i	−0.995728	+	0.0923398i	$0.970565\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.69364i	0.224328i
$58$	0	0
$59$	−4.38793	−0.571260	−0.285630	−	0.958340i	$-0.592203\pi$
$59$	−4.38793	−0.571260	−0.285630	+	0.958340i	$0.592203\pi$
$60$	0	0
$61$	−15.3316	−1.96300	−0.981502	−	0.191452i	$-0.938681\pi$
$61$	−15.3316	−1.96300	−0.981502	+	0.191452i	$0.938681\pi$
$62$	0	0
$63$	1.74037i	0.219266i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.23075i	1.12772i	0.825872	+	0.563858i	$0.190683\pi$
$67$	9.23075i	1.12772i	−0.825872	+	0.563858i	$0.809317\pi$
$68$	0	0
$69$	8.67867	1.04479
$70$	0	0
$71$	0.0235645	0.00279659	0.00139830	−	0.999999i	$-0.499555\pi$
$71$	0.0235645	0.00279659	0.00139830	+	0.999999i	$0.499555\pi$
$72$	0	0
$73$	1.02251i	0.119676i	0.998208	+	0.0598380i	$0.0190584\pi$
$73$	1.02251i	0.119676i	−0.998208	+	0.0598380i	$0.980942\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.03813i	− 0.460187i
$78$	0	0
$79$	0.798776	0.0898693	0.0449346	−	0.998990i	$-0.485692\pi$
$79$	0.798776	0.0898693	0.0449346	+	0.998990i	$0.485692\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 2.37008i	− 0.260150i	−0.991504	−	0.130075i	$-0.958478\pi$
$83$	− 2.37008i	− 0.260150i	0.991504	−	0.130075i	$-0.0415219\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 3.12233i	− 0.334749i
$88$	0	0
$89$	4.86205	0.515376	0.257688	−	0.966228i	$-0.417039\pi$
$89$	4.86205	0.515376	0.257688	+	0.966228i	$0.417039\pi$
$90$	0	0
$91$	−8.59447	−0.900945
$92$	0	0
$93$	9.56166i	0.991498i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 3.46151i	− 0.351463i	−0.984438	−	0.175731i	$-0.943771\pi$
$97$	− 3.46151i	− 0.351463i	0.984438	−	0.175731i	$-0.0562290\pi$
$98$	0	0
$99$	−2.32027	−0.233196
$100$	0	0
$101$	−7.58056	−0.754294	−0.377147	−	0.926154i	$-0.623095\pi$
$101$	−7.58056	−0.754294	−0.377147	+	0.926154i	$0.623095\pi$
$102$	0	0
$103$	− 10.7699i	− 1.06119i	−0.847626	−	0.530595i	$-0.821969\pi$
$103$	− 10.7699i	− 1.06119i	0.847626	−	0.530595i	$-0.178031\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.34344i	− 0.419896i	−0.977713	−	0.209948i	$-0.932671\pi$
$107$	− 4.34344i	− 0.419896i	0.977713	−	0.209948i	$-0.0673294\pi$
$108$	0	0
$109$	−1.07127	−0.102609	−0.0513045	−	0.998683i	$-0.516338\pi$
$109$	−1.07127	−0.102609	−0.0513045	+	0.998683i	$0.516338\pi$
$110$	0	0
$111$	−5.36700	−0.509413
$112$	0	0
$113$	− 18.5617i	− 1.74613i	−0.487601	−	0.873067i	$-0.662128\pi$
$113$	− 18.5617i	− 1.74613i	0.487601	−	0.873067i	$-0.337872\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.93831i	0.456547i
$118$	0	0
$119$	6.50961	0.596735
$120$	0	0
$121$	−5.61633	−0.510576
$122$	0	0
$123$	− 8.72540i	− 0.786743i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	19.0096i	1.68683i	0.537265	+	0.843414i	$0.319458\pi$
$127$	19.0096i	1.68683i	−0.537265	+	0.843414i	$0.680542\pi$
$128$	0	0
$129$	2.86270	0.252047
$130$	0	0
$131$	−0.745032	−0.0650937	−0.0325469	−	0.999470i	$-0.510362\pi$
$131$	−0.745032	−0.0650937	−0.0325469	+	0.999470i	$0.510362\pi$
$132$	0	0
$133$	− 2.94756i	− 0.255586i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 11.1326i	− 0.951119i	−0.879684	−	0.475559i	$-0.842246\pi$
$137$	− 11.1326i	− 0.951119i	0.879684	−	0.475559i	$-0.157754\pi$
$138$	0	0
$139$	−13.6889	−1.16108	−0.580539	−	0.814233i	$-0.697158\pi$
$139$	−13.6889	−1.16108	−0.580539	+	0.814233i	$0.697158\pi$
$140$	0	0
$141$	8.46248	0.712670
$142$	0	0
$143$	− 11.4582i	− 0.958185i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.97112i	0.327533i
$148$	0	0
$149$	10.6283	0.870701	0.435351	−	0.900261i	$-0.356624\pi$
$149$	10.6283	0.870701	0.435351	+	0.900261i	$0.356624\pi$
$150$	0	0
$151$	−10.4689	−0.851943	−0.425972	−	0.904737i	$-0.640068\pi$
$151$	−10.4689	−0.851943	−0.425972	+	0.904737i	$0.640068\pi$
$152$	0	0
$153$	− 3.74037i	− 0.302391i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 17.2987i	− 1.38059i	−0.723528	−	0.690295i	$-0.757481\pi$
$157$	− 17.2987i	− 1.38059i	0.723528	−	0.690295i	$-0.242519\pi$
$158$	0	0
$159$	−1.34449	−0.106625
$160$	0	0
$161$	−15.1041	−1.19037
$162$	0	0
$163$	6.14688i	0.481460i	0.970592	+	0.240730i	$0.0773869\pi$
$163$	6.14688i	0.481460i	−0.970592	+	0.240730i	$0.922613\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1.90018i	− 0.147040i	−0.997294	−	0.0735201i	$-0.976577\pi$
$167$	− 1.90018i	− 0.147040i	0.997294	−	0.0735201i	$-0.0234233\pi$
$168$	0	0
$169$	−11.3869	−0.875914
$170$	0	0
$171$	−1.69364	−0.129516
$172$	0	0
$173$	− 11.5859i	− 0.880857i	−0.897788	−	0.440429i	$-0.854826\pi$
$173$	− 11.5859i	− 0.880857i	0.897788	−	0.440429i	$-0.145174\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 4.38793i	− 0.329817i
$178$	0	0
$179$	−22.0477	−1.64792	−0.823961	−	0.566646i	$-0.808241\pi$
$179$	−22.0477	−1.64792	−0.823961	+	0.566646i	$0.808241\pi$
$180$	0	0
$181$	20.6571	1.53543	0.767717	−	0.640790i	$-0.221393\pi$
$181$	20.6571	1.53543	0.767717	+	0.640790i	$0.221393\pi$
$182$	0	0
$183$	− 15.3316i	− 1.13334i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	8.67867i	0.634648i
$188$	0	0
$189$	−1.74037	−0.126593
$190$	0	0
$191$	20.8321	1.50736	0.753680	−	0.657242i	$-0.228277\pi$
$191$	20.8321	1.50736	0.753680	+	0.657242i	$0.228277\pi$
$192$	0	0
$193$	− 4.78053i	− 0.344110i	−0.985087	−	0.172055i	$-0.944959\pi$
$193$	− 4.78053i	− 0.344110i	0.985087	−	0.172055i	$-0.0550406\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.3520i	1.23628i	0.786068	+	0.618141i	$0.212114\pi$
$197$	17.3520i	1.23628i	−0.786068	+	0.618141i	$0.787886\pi$
$198$	0	0
$199$	−11.0756	−0.785129	−0.392564	−	0.919724i	$-0.628412\pi$
$199$	−11.0756	−0.785129	−0.392564	+	0.919724i	$0.628412\pi$
$200$	0	0
$201$	−9.23075	−0.651087
$202$	0	0
$203$	5.43401i	0.381393i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.67867i	0.603210i
$208$	0	0
$209$	3.92971	0.271824
$210$	0	0
$211$	5.66902	0.390272	0.195136	−	0.980776i	$-0.437485\pi$
$211$	5.66902	0.390272	0.195136	+	0.980776i	$0.437485\pi$
$212$	0	0
$213$	0.0235645i	0.00161461i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 16.6408i	− 1.12965i
$218$	0	0
$219$	−1.02251	−0.0690950
$220$	0	0
$221$	18.4711	1.24250
$222$	0	0
$223$	21.3805i	1.43174i	0.698231	+	0.715872i	$0.253971\pi$
$223$	21.3805i	1.43174i	−0.698231	+	0.715872i	$0.746029\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 22.9624i	− 1.52407i	−0.647535	−	0.762036i	$-0.724200\pi$
$227$	− 22.9624i	− 1.52407i	0.647535	−	0.762036i	$-0.275800\pi$
$228$	0	0
$229$	28.1047	1.85721	0.928604	−	0.371072i	$-0.121010\pi$
$229$	28.1047	1.85721	0.928604	+	0.371072i	$0.121010\pi$
$230$	0	0
$231$	4.03813	0.265689
$232$	0	0
$233$	1.33195i	0.0872592i	0.999048	+	0.0436296i	$0.0138921\pi$
$233$	1.33195i	0.0872592i	−0.999048	+	0.0436296i	$0.986108\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0.798776i	0.0518861i
$238$	0	0
$239$	−3.44529	−0.222857	−0.111429	−	0.993772i	$-0.535543\pi$
$239$	−3.44529	−0.222857	−0.111429	+	0.993772i	$0.535543\pi$
$240$	0	0
$241$	−24.0938	−1.55202	−0.776008	−	0.630722i	$-0.782759\pi$
$241$	−24.0938	−1.55202	−0.776008	+	0.630722i	$0.782759\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 8.36372i	− 0.532171i
$248$	0	0
$249$	2.37008	0.150198
$250$	0	0
$251$	−30.1621	−1.90381	−0.951907	−	0.306387i	$-0.900880\pi$
$251$	−30.1621	−1.90381	−0.951907	+	0.306387i	$0.900880\pi$
$252$	0	0
$253$	− 20.1369i	− 1.26600i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.6862i	0.666588i	0.942823	+	0.333294i	$0.108160\pi$
$257$	10.6862i	0.666588i	−0.942823	+	0.333294i	$0.891840\pi$
$258$	0	0
$259$	9.34055	0.580394
$260$	0	0
$261$	3.12233	0.193268
$262$	0	0
$263$	− 9.23278i	− 0.569318i	−0.958629	−	0.284659i	$-0.908120\pi$
$263$	− 9.23278i	− 0.569318i	0.958629	−	0.284659i	$-0.0918803\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.86205i	0.297553i
$268$	0	0
$269$	−26.7381	−1.63025	−0.815124	−	0.579286i	$-0.803331\pi$
$269$	−26.7381	−1.63025	−0.815124	+	0.579286i	$0.803331\pi$
$270$	0	0
$271$	24.1840	1.46907	0.734535	−	0.678571i	$-0.237400\pi$
$271$	24.1840	1.46907	0.734535	+	0.678571i	$0.237400\pi$
$272$	0	0
$273$	− 8.59447i	− 0.520161i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 31.6079i	− 1.89914i	−0.313563	−	0.949568i	$-0.601523\pi$
$277$	− 31.6079i	− 1.89914i	0.313563	−	0.949568i	$-0.398477\pi$
$278$	0	0
$279$	−9.56166	−0.572441
$280$	0	0
$281$	23.2625	1.38773	0.693863	−	0.720107i	$-0.255907\pi$
$281$	23.2625	1.38773	0.693863	+	0.720107i	$0.255907\pi$
$282$	0	0
$283$	− 24.8321i	− 1.47612i	−0.674737	−	0.738058i	$-0.735743\pi$
$283$	− 24.8321i	− 1.47612i	0.674737	−	0.738058i	$-0.264257\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.1854i	0.896366i
$288$	0	0
$289$	3.00965	0.177038
$290$	0	0
$291$	3.46151	0.202917
$292$	0	0
$293$	− 29.2758i	− 1.71031i	−0.518371	−	0.855156i	$-0.673461\pi$
$293$	− 29.2758i	− 1.71031i	0.518371	−	0.855156i	$-0.326539\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.32027i	− 0.134636i
$298$	0	0
$299$	−42.8580	−2.47854
$300$	0	0
$301$	−4.98215	−0.287166
$302$	0	0
$303$	− 7.58056i	− 0.435492i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.73972i	0.498802i	0.968400	+	0.249401i	$0.0802337\pi$
$307$	8.73972i	0.498802i	−0.968400	+	0.249401i	$0.919766\pi$
$308$	0	0
$309$	10.7699	0.612678
$310$	0	0
$311$	−15.4561	−0.876436	−0.438218	−	0.898869i	$-0.644390\pi$
$311$	−15.4561	−0.876436	−0.438218	+	0.898869i	$0.644390\pi$
$312$	0	0
$313$	1.55898i	0.0881185i	0.999029	+	0.0440593i	$0.0140290\pi$
$313$	1.55898i	0.0881185i	−0.999029	+	0.0440593i	$0.985971\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 5.46256i	− 0.306808i	−0.988164	−	0.153404i	$-0.950976\pi$
$317$	− 5.46256i	− 0.306808i	0.988164	−	0.153404i	$-0.0490236\pi$
$318$	0	0
$319$	−7.24467	−0.405623
$320$	0	0
$321$	4.34344	0.242427
$322$	0	0
$323$	6.33484i	0.352480i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 1.07127i	− 0.0592414i
$328$	0	0
$329$	−14.7278	−0.811972
$330$	0	0
$331$	0.855804	0.0470392	0.0235196	−	0.999723i	$-0.492513\pi$
$331$	0.855804	0.0470392	0.0235196	+	0.999723i	$0.492513\pi$
$332$	0	0
$333$	− 5.36700i	− 0.294110i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	14.8246i	0.807546i	0.914859	+	0.403773i	$0.132301\pi$
$337$	14.8246i	0.807546i	−0.914859	+	0.403773i	$0.867699\pi$
$338$	0	0
$339$	18.5617	1.00813
$340$	0	0
$341$	22.1857	1.20142
$342$	0	0
$343$	− 19.0938i	− 1.03097i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.17069i	− 0.0628459i	−0.999506	−	0.0314229i	$-0.989996\pi$
$347$	− 1.17069i	− 0.0628459i	0.999506	−	0.0314229i	$-0.0100039\pi$
$348$	0	0
$349$	21.4346	1.14737	0.573683	−	0.819077i	$-0.305514\pi$
$349$	21.4346	1.14737	0.573683	+	0.819077i	$0.305514\pi$
$350$	0	0
$351$	−4.93831	−0.263587
$352$	0	0
$353$	3.56271i	0.189624i	0.995495	+	0.0948119i	$0.0302250\pi$
$353$	3.56271i	0.189624i	−0.995495	+	0.0948119i	$0.969775\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.50961i	0.344525i
$358$	0	0
$359$	−13.3912	−0.706761	−0.353381	−	0.935480i	$-0.614968\pi$
$359$	−13.3912	−0.706761	−0.353381	+	0.935480i	$0.614968\pi$
$360$	0	0
$361$	−16.1316	−0.849031
$362$	0	0
$363$	− 5.61633i	− 0.294781i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1.42436i	− 0.0743508i	−0.999309	−	0.0371754i	$-0.988164\pi$
$367$	− 1.42436i	− 0.0743508i	0.999309	−	0.0371754i	$-0.0118360\pi$
$368$	0	0
$369$	8.72540	0.454226
$370$	0	0
$371$	2.33990	0.121482
$372$	0	0
$373$	− 25.5327i	− 1.32203i	−0.750371	−	0.661017i	$-0.770125\pi$
$373$	− 25.5327i	− 1.32203i	0.750371	−	0.661017i	$-0.229875\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	15.4190i	0.794121i
$378$	0	0
$379$	38.0791	1.95599	0.977995	−	0.208628i	$-0.0668997\pi$
$379$	38.0791	1.95599	0.977995	+	0.208628i	$0.0668997\pi$
$380$	0	0
$381$	−19.0096	−0.973890
$382$	0	0
$383$	− 13.6545i	− 0.697710i	−0.937177	−	0.348855i	$-0.886571\pi$
$383$	− 13.6545i	− 0.697710i	0.937177	−	0.348855i	$-0.113429\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.86270i	0.145519i
$388$	0	0
$389$	−35.7899	−1.81462	−0.907310	−	0.420462i	$-0.861868\pi$
$389$	−35.7899	−1.81462	−0.907310	+	0.420462i	$0.861868\pi$
$390$	0	0
$391$	32.4614	1.64165
$392$	0	0
$393$	− 0.745032i	− 0.0375819i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	24.5424i	1.23175i	0.787846	+	0.615873i	$0.211196\pi$
$397$	24.5424i	1.23175i	−0.787846	+	0.615873i	$0.788804\pi$
$398$	0	0
$399$	2.94756	0.147562
$400$	0	0
$401$	−25.7068	−1.28373	−0.641867	−	0.766816i	$-0.721840\pi$
$401$	−25.7068	−1.28373	−0.641867	+	0.766816i	$0.721840\pi$
$402$	0	0
$403$	− 47.2184i	− 2.35212i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12.4529i	0.617268i
$408$	0	0
$409$	18.5559	0.917532	0.458766	−	0.888557i	$-0.348292\pi$
$409$	18.5559	0.917532	0.458766	+	0.888557i	$0.348292\pi$
$410$	0	0
$411$	11.1326	0.549129
$412$	0	0
$413$	7.63661i	0.375773i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 13.6889i	− 0.670348i
$418$	0	0
$419$	−4.69324	−0.229280	−0.114640	−	0.993407i	$-0.536571\pi$
$419$	−4.69324	−0.229280	−0.114640	+	0.993407i	$0.536571\pi$
$420$	0	0
$421$	9.22050	0.449380	0.224690	−	0.974430i	$-0.427863\pi$
$421$	9.22050	0.449380	0.224690	+	0.974430i	$0.427863\pi$
$422$	0	0
$423$	8.46248i	0.411460i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	26.6825i	1.29126i
$428$	0	0
$429$	11.4582	0.553208
$430$	0	0
$431$	28.5809	1.37669	0.688346	−	0.725382i	$-0.258337\pi$
$431$	28.5809	1.37669	0.688346	+	0.725382i	$0.258337\pi$
$432$	0	0
$433$	− 3.63365i	− 0.174622i	−0.996181	−	0.0873110i	$-0.972173\pi$
$433$	− 3.63365i	− 0.174622i	0.996181	−	0.0873110i	$-0.0278274\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 14.6986i	− 0.703127i
$438$	0	0
$439$	17.6073	0.840352	0.420176	−	0.907443i	$-0.361968\pi$
$439$	17.6073	0.840352	0.420176	+	0.907443i	$0.361968\pi$
$440$	0	0
$441$	−3.97112	−0.189101
$442$	0	0
$443$	− 7.11807i	− 0.338190i	−0.985600	−	0.169095i	$-0.945916\pi$
$443$	− 7.11807i	− 0.338190i	0.985600	−	0.169095i	$-0.0540844\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	10.6283i	0.502700i
$448$	0	0
$449$	20.4121	0.963305	0.481653	−	0.876362i	$-0.340037\pi$
$449$	20.4121	0.963305	0.481653	+	0.876362i	$0.340037\pi$
$450$	0	0
$451$	−20.2453	−0.953315
$452$	0	0
$453$	− 10.4689i	− 0.491870i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 16.8375i	− 0.787624i	−0.919191	−	0.393812i	$-0.871156\pi$
$457$	− 16.8375i	− 0.787624i	0.919191	−	0.393812i	$-0.128844\pi$
$458$	0	0
$459$	3.74037	0.174585
$460$	0	0
$461$	17.3249	0.806903	0.403451	−	0.915001i	$-0.367810\pi$
$461$	17.3249	0.806903	0.403451	+	0.915001i	$0.367810\pi$
$462$	0	0
$463$	− 8.60412i	− 0.399867i	−0.979809	−	0.199934i	$-0.935927\pi$
$463$	− 8.60412i	− 0.399867i	0.979809	−	0.199934i	$-0.0640727\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0.835852i	0.0386786i	0.999813	+	0.0193393i	$0.00615628\pi$
$467$	0.835852i	0.0386786i	−0.999813	+	0.0193393i	$0.993844\pi$
$468$	0	0
$469$	16.0649	0.741808
$470$	0	0
$471$	17.2987	0.797084
$472$	0	0
$473$	− 6.64225i	− 0.305411i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1.34449i	− 0.0615599i
$478$	0	0
$479$	33.4495	1.52835	0.764173	−	0.645012i	$-0.223147\pi$
$479$	33.4495	1.52835	0.764173	+	0.645012i	$0.223147\pi$
$480$	0	0
$481$	26.5039	1.20847
$482$	0	0
$483$	− 15.1041i	− 0.687260i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 24.4158i	− 1.10638i	−0.833054	−	0.553192i	$-0.813410\pi$
$487$	− 24.4158i	− 1.10638i	0.833054	−	0.553192i	$-0.186590\pi$
$488$	0	0
$489$	−6.14688	−0.277971
$490$	0	0
$491$	−14.6475	−0.661032	−0.330516	−	0.943800i	$-0.607223\pi$
$491$	−14.6475	−0.661032	−0.330516	+	0.943800i	$0.607223\pi$
$492$	0	0
$493$	− 11.6787i	− 0.525981i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 0.0410109i	− 0.00183959i
$498$	0	0
$499$	13.5655	0.607276	0.303638	−	0.952787i	$-0.401799\pi$
$499$	13.5655	0.607276	0.303638	+	0.952787i	$0.401799\pi$
$500$	0	0
$501$	1.90018	0.0848937
$502$	0	0
$503$	40.9282i	1.82490i	0.409191	+	0.912449i	$0.365811\pi$
$503$	40.9282i	1.82490i	−0.409191	+	0.912449i	$0.634189\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 11.3869i	− 0.505709i
$508$	0	0
$509$	−12.1412	−0.538147	−0.269074	−	0.963120i	$-0.586718\pi$
$509$	−12.1412	−0.538147	−0.269074	+	0.963120i	$0.586718\pi$
$510$	0	0
$511$	1.77955	0.0787226
$512$	0	0
$513$	− 1.69364i	− 0.0747760i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 19.6353i	− 0.863559i
$518$	0	0
$519$	11.5859	0.508563
$520$	0	0
$521$	26.2250	1.14894	0.574470	−	0.818526i	$-0.305208\pi$
$521$	26.2250	1.14894	0.574470	+	0.818526i	$0.305208\pi$
$522$	0	0
$523$	− 15.3245i	− 0.670095i	−0.942201	−	0.335048i	$-0.891248\pi$
$523$	− 15.3245i	− 0.670095i	0.942201	−	0.335048i	$-0.108752\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	35.7641i	1.55791i
$528$	0	0
$529$	−52.3194	−2.27476
$530$	0	0
$531$	4.38793	0.190420
$532$	0	0
$533$	43.0887i	1.86638i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 22.0477i	− 0.951429i
$538$	0	0
$539$	9.21409	0.396879
$540$	0	0
$541$	9.47745	0.407467	0.203734	−	0.979026i	$-0.434692\pi$
$541$	9.47745	0.407467	0.203734	+	0.979026i	$0.434692\pi$
$542$	0	0
$543$	20.6571i	0.886483i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0.645935i	0.0276182i	0.999905	+	0.0138091i	$0.00439571\pi$
$547$	0.645935i	0.0276182i	−0.999905	+	0.0138091i	$0.995604\pi$
$548$	0	0
$549$	15.3316	0.654335
$550$	0	0
$551$	−5.28811	−0.225281
$552$	0	0
$553$	− 1.39016i	− 0.0591158i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 12.0297i	− 0.509716i	−0.966978	−	0.254858i	$-0.917971\pi$
$557$	− 12.0297i	− 0.509716i	0.966978	−	0.254858i	$-0.0820287\pi$
$558$	0	0
$559$	−14.1369	−0.597927
$560$	0	0
$561$	−8.67867	−0.366414
$562$	0	0
$563$	− 15.8832i	− 0.669398i	−0.942325	−	0.334699i	$-0.891365\pi$
$563$	− 15.8832i	− 0.669398i	0.942325	−	0.334699i	$-0.108635\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.74037i	− 0.0730886i
$568$	0	0
$569$	−21.9304	−0.919368	−0.459684	−	0.888082i	$-0.652037\pi$
$569$	−21.9304	−0.919368	−0.459684	+	0.888082i	$0.652037\pi$
$570$	0	0
$571$	−6.80835	−0.284921	−0.142460	−	0.989801i	$-0.545501\pi$
$571$	−6.80835	−0.284921	−0.142460	+	0.989801i	$0.545501\pi$
$572$	0	0
$573$	20.8321i	0.870274i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	18.6273i	0.775464i	0.921772	+	0.387732i	$0.126741\pi$
$577$	18.6273i	0.775464i	−0.921772	+	0.387732i	$0.873259\pi$
$578$	0	0
$579$	4.78053	0.198672
$580$	0	0
$581$	−4.12481	−0.171126
$582$	0	0
$583$	3.11958i	0.129200i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 14.2102i	− 0.586518i	−0.956033	−	0.293259i	$-0.905260\pi$
$587$	− 14.2102i	− 0.586518i	0.956033	−	0.293259i	$-0.0947398\pi$
$588$	0	0
$589$	16.1940	0.667262
$590$	0	0
$591$	−17.3520	−0.713767
$592$	0	0
$593$	− 40.7850i	− 1.67484i	−0.546559	−	0.837420i	$-0.684063\pi$
$593$	− 40.7850i	− 1.67484i	0.546559	−	0.837420i	$-0.315937\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 11.0756i	− 0.453294i
$598$	0	0
$599$	−34.8928	−1.42568	−0.712841	−	0.701326i	$-0.752592\pi$
$599$	−34.8928	−1.42568	−0.712841	+	0.701326i	$0.752592\pi$
$600$	0	0
$601$	10.6287	0.433552	0.216776	−	0.976221i	$-0.430446\pi$
$601$	10.6287	0.433552	0.216776	+	0.976221i	$0.430446\pi$
$602$	0	0
$603$	− 9.23075i	− 0.375905i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	30.5033i	1.23809i	0.785355	+	0.619045i	$0.212480\pi$
$607$	30.5033i	1.23809i	−0.785355	+	0.619045i	$0.787520\pi$
$608$	0	0
$609$	−5.43401	−0.220197
$610$	0	0
$611$	−41.7904	−1.69066
$612$	0	0
$613$	12.7732i	0.515904i	0.966158	+	0.257952i	$0.0830476\pi$
$613$	12.7732i	0.515904i	−0.966158	+	0.257952i	$0.916952\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 11.9327i	− 0.480391i	−0.970725	−	0.240196i	$-0.922788\pi$
$617$	− 11.9327i	− 0.480391i	0.970725	−	0.240196i	$-0.0772116\pi$
$618$	0	0
$619$	14.5045	0.582987	0.291494	−	0.956573i	$-0.405848\pi$
$619$	14.5045	0.582987	0.291494	+	0.956573i	$0.405848\pi$
$620$	0	0
$621$	−8.67867	−0.348263
$622$	0	0
$623$	− 8.46176i	− 0.339013i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.92971i	0.156937i
$628$	0	0
$629$	−20.0746	−0.800425
$630$	0	0
$631$	−3.66780	−0.146013	−0.0730064	−	0.997331i	$-0.523259\pi$
$631$	−3.66780	−0.146013	−0.0730064	+	0.997331i	$0.523259\pi$
$632$	0	0
$633$	5.66902i	0.225323i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 19.6106i	− 0.777001i
$638$	0	0
$639$	−0.0235645	−0.000932198 0
$640$	0	0
$641$	−13.8519	−0.547116	−0.273558	−	0.961856i	$-0.588201\pi$
$641$	−13.8519	−0.547116	−0.273558	+	0.961856i	$0.588201\pi$
$642$	0	0
$643$	27.1641i	1.07125i	0.844456	+	0.535624i	$0.179924\pi$
$643$	27.1641i	1.07125i	−0.844456	+	0.535624i	$0.820076\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 5.14988i	− 0.202463i	−0.994863	−	0.101231i	$-0.967722\pi$
$647$	− 5.14988i	− 0.202463i	0.994863	−	0.101231i	$-0.0322782\pi$
$648$	0	0
$649$	−10.1812	−0.399647
$650$	0	0
$651$	16.6408	0.652204
$652$	0	0
$653$	29.6734i	1.16121i	0.814186	+	0.580604i	$0.197184\pi$
$653$	29.6734i	1.16121i	−0.814186	+	0.580604i	$0.802816\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 1.02251i	− 0.0398920i
$658$	0	0
$659$	−15.7740	−0.614466	−0.307233	−	0.951634i	$-0.599403\pi$
$659$	−15.7740	−0.614466	−0.307233	+	0.951634i	$0.599403\pi$
$660$	0	0
$661$	−24.0557	−0.935657	−0.467828	−	0.883819i	$-0.654963\pi$
$661$	−24.0557	−0.935657	−0.467828	+	0.883819i	$0.654963\pi$
$662$	0	0
$663$	18.4711i	0.717357i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	27.0977i	1.04923i
$668$	0	0
$669$	−21.3805	−0.826618
$670$	0	0
$671$	−35.5734	−1.37330
$672$	0	0
$673$	− 32.0035i	− 1.23365i	−0.787102	−	0.616823i	$-0.788420\pi$
$673$	− 32.0035i	− 1.23365i	0.787102	−	0.616823i	$-0.211580\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 24.3809i	− 0.937035i	−0.883454	−	0.468517i	$-0.844788\pi$
$677$	− 24.3809i	− 0.937035i	0.883454	−	0.468517i	$-0.155212\pi$
$678$	0	0
$679$	−6.02429	−0.231191
$680$	0	0
$681$	22.9624	0.879923
$682$	0	0
$683$	11.1528i	0.426752i	0.976970	+	0.213376i	$0.0684459\pi$
$683$	11.1528i	0.426752i	−0.976970	+	0.213376i	$0.931554\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	28.1047i	1.07226i
$688$	0	0
$689$	6.63949	0.252945
$690$	0	0
$691$	10.8286	0.411941	0.205970	−	0.978558i	$-0.433965\pi$
$691$	10.8286	0.411941	0.205970	+	0.978558i	$0.433965\pi$
$692$	0	0
$693$	4.03813i	0.153396i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 32.6362i	− 1.23618i
$698$	0	0
$699$	−1.33195	−0.0503791
$700$	0	0
$701$	43.2512	1.63357	0.816787	−	0.576939i	$-0.195753\pi$
$701$	43.2512	1.63357	0.816787	+	0.576939i	$0.195753\pi$
$702$	0	0
$703$	9.08977i	0.342827i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.1930i	0.496172i
$708$	0	0
$709$	44.6052	1.67518	0.837592	−	0.546296i	$-0.183963\pi$
$709$	44.6052	1.67518	0.837592	+	0.546296i	$0.183963\pi$
$710$	0	0
$711$	−0.798776	−0.0299564
$712$	0	0
$713$	− 82.9825i	− 3.10772i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 3.44529i	− 0.128667i
$718$	0	0
$719$	32.0868	1.19664	0.598318	−	0.801259i	$-0.295836\pi$
$719$	32.0868	1.19664	0.598318	+	0.801259i	$0.295836\pi$
$720$	0	0
$721$	−18.7436	−0.698047
$722$	0	0
$723$	− 24.0938i	− 0.896057i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 13.6216i	− 0.505196i	−0.967571	−	0.252598i	$-0.918715\pi$
$727$	− 13.6216i	− 0.505196i	0.967571	−	0.252598i	$-0.0812850\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	10.7076	0.396033
$732$	0	0
$733$	− 28.4242i	− 1.04987i	−0.851142	−	0.524935i	$-0.824090\pi$
$733$	− 28.4242i	− 1.04987i	0.851142	−	0.524935i	$-0.175910\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21.4179i	0.788937i
$738$	0	0
$739$	−9.71457	−0.357356	−0.178678	−	0.983908i	$-0.557182\pi$
$739$	−9.71457	−0.357356	−0.178678	+	0.983908i	$0.557182\pi$
$740$	0	0
$741$	8.36372	0.307249
$742$	0	0
$743$	− 34.2029i	− 1.25478i	−0.778705	−	0.627390i	$-0.784123\pi$
$743$	− 34.2029i	− 1.25478i	0.778705	−	0.627390i	$-0.215877\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.37008i	0.0867168i
$748$	0	0
$749$	−7.55917	−0.276206
$750$	0	0
$751$	−12.7840	−0.466495	−0.233248	−	0.972417i	$-0.574935\pi$
$751$	−12.7840	−0.466495	−0.233248	+	0.972417i	$0.574935\pi$
$752$	0	0
$753$	− 30.1621i	− 1.09917i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	26.6200i	0.967520i	0.875201	+	0.483760i	$0.160729\pi$
$757$	26.6200i	0.967520i	−0.875201	+	0.483760i	$0.839271\pi$
$758$	0	0
$759$	20.1369	0.730923
$760$	0	0
$761$	25.7137	0.932122	0.466061	−	0.884753i	$-0.345673\pi$
$761$	25.7137	0.932122	0.466061	+	0.884753i	$0.345673\pi$
$762$	0	0
$763$	1.86440i	0.0674959i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	21.6689i	0.782420i
$768$	0	0
$769$	38.3717	1.38372	0.691860	−	0.722032i	$-0.256792\pi$
$769$	38.3717	1.38372	0.691860	+	0.722032i	$0.256792\pi$
$770$	0	0
$771$	−10.6862	−0.384855
$772$	0	0
$773$	1.39884i	0.0503126i	0.999684	+	0.0251563i	$0.00800835\pi$
$773$	1.39884i	0.0503126i	−0.999684	+	0.0251563i	$0.991992\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	9.34055i	0.335090i
$778$	0	0
$779$	−14.7777	−0.529466
$780$	0	0
$781$	0.0546761	0.00195647
$782$	0	0
$783$	3.12233i	0.111583i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 4.21023i	− 0.150078i	−0.997181	−	0.0750392i	$-0.976092\pi$
$787$	− 4.21023i	− 0.150078i	0.997181	−	0.0750392i	$-0.0239082\pi$
$788$	0	0
$789$	9.23278	0.328696
$790$	0	0
$791$	−32.3041	−1.14860
$792$	0	0
$793$	75.7119i	2.68861i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 31.6196i	− 1.12002i	−0.828485	−	0.560012i	$-0.810797\pi$
$797$	− 31.6196i	− 1.12002i	0.828485	−	0.560012i	$-0.189203\pi$
$798$	0	0
$799$	31.6528	1.11980
$800$	0	0
$801$	−4.86205	−0.171792
$802$	0	0
$803$	2.37251i	0.0837240i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 26.7381i	− 0.941224i
$808$	0	0
$809$	−42.1098	−1.48050	−0.740252	−	0.672330i	$-0.765294\pi$
$809$	−42.1098	−1.48050	−0.740252	+	0.672330i	$0.765294\pi$
$810$	0	0
$811$	15.1530	0.532095	0.266048	−	0.963960i	$-0.414282\pi$
$811$	15.1530	0.532095	0.266048	+	0.963960i	$0.414282\pi$
$812$	0	0
$813$	24.1840i	0.848168i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 4.84839i	− 0.169624i
$818$	0	0
$819$	8.59447	0.300315
$820$	0	0
$821$	−0.402320	−0.0140411	−0.00702053	−	0.999975i	$-0.502235\pi$
$821$	−0.402320	−0.0140411	−0.00702053	+	0.999975i	$0.502235\pi$
$822$	0	0
$823$	0.609361i	0.0212410i	0.999944	+	0.0106205i	$0.00338067\pi$
$823$	0.609361i	0.0212410i	−0.999944	+	0.0106205i	$0.996619\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27.4672i	0.955128i	0.878597	+	0.477564i	$0.158480\pi$
$827$	27.4672i	0.955128i	−0.878597	+	0.477564i	$0.841520\pi$
$828$	0	0
$829$	−31.1543	−1.08204	−0.541018	−	0.841011i	$-0.681961\pi$
$829$	−31.1543	−1.08204	−0.541018	+	0.841011i	$0.681961\pi$
$830$	0	0
$831$	31.6079	1.09647
$832$	0	0
$833$	14.8535i	0.514642i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 9.56166i	− 0.330499i
$838$	0	0
$839$	−39.1176	−1.35049	−0.675245	−	0.737594i	$-0.735962\pi$
$839$	−39.1176	−1.35049	−0.675245	+	0.737594i	$0.735962\pi$
$840$	0	0
$841$	−19.2510	−0.663829
$842$	0	0
$843$	23.2625i	0.801204i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	9.77448i	0.335855i
$848$	0	0
$849$	24.8321	0.852236
$850$	0	0
$851$	46.5785	1.59669
$852$	0	0
$853$	52.5757i	1.80016i	0.435727	+	0.900079i	$0.356491\pi$
$853$	52.5757i	1.80016i	−0.435727	+	0.900079i	$0.643509\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 31.0291i	− 1.05993i	−0.848018	−	0.529967i	$-0.822204\pi$
$857$	− 31.0291i	− 1.05993i	0.848018	−	0.529967i	$-0.177796\pi$
$858$	0	0
$859$	−38.8450	−1.32537	−0.662687	−	0.748897i	$-0.730584\pi$
$859$	−38.8450	−1.32537	−0.662687	+	0.748897i	$0.730584\pi$
$860$	0	0
$861$	−15.1854	−0.517517
$862$	0	0
$863$	3.60246i	0.122629i	0.998118	+	0.0613147i	$0.0195293\pi$
$863$	3.60246i	0.122629i	−0.998118	+	0.0613147i	$0.980471\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.00965i	0.102213i
$868$	0	0
$869$	1.85338	0.0628715
$870$	0	0
$871$	45.5843	1.54456
$872$	0	0
$873$	3.46151i	0.117154i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 31.5597i	− 1.06569i	−0.846211	−	0.532847i	$-0.821122\pi$
$877$	− 31.5597i	− 1.06569i	0.846211	−	0.532847i	$-0.178878\pi$
$878$	0	0
$879$	29.2758	0.987449
$880$	0	0
$881$	6.02783	0.203083	0.101541	−	0.994831i	$-0.467623\pi$
$881$	6.02783	0.203083	0.101541	+	0.994831i	$0.467623\pi$
$882$	0	0
$883$	3.96581i	0.133460i	0.997771	+	0.0667300i	$0.0212566\pi$
$883$	3.96581i	0.133460i	−0.997771	+	0.0667300i	$0.978743\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	55.7151i	1.87073i	0.353684	+	0.935365i	$0.384929\pi$
$887$	55.7151i	1.87073i	−0.353684	+	0.935365i	$0.615071\pi$
$888$	0	0
$889$	33.0837	1.10959
$890$	0	0
$891$	2.32027	0.0777321
$892$	0	0
$893$	− 14.3324i	− 0.479616i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 42.8580i	− 1.43099i
$898$	0	0
$899$	−29.8547	−0.995709
$900$	0	0
$901$	−5.02888	−0.167536
$902$	0	0
$903$	− 4.98215i	− 0.165796i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 21.4063i	− 0.710785i	−0.934717	−	0.355392i	$-0.884347\pi$
$907$	− 21.4063i	− 0.710785i	0.934717	−	0.355392i	$-0.115653\pi$
$908$	0	0
$909$	7.58056	0.251431
$910$	0	0
$911$	31.5890	1.04659	0.523295	−	0.852151i	$-0.324702\pi$
$911$	31.5890	1.04659	0.523295	+	0.852151i	$0.324702\pi$
$912$	0	0
$913$	− 5.49924i	− 0.181998i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.29663i	0.0428185i
$918$	0	0
$919$	18.0775	0.596322	0.298161	−	0.954516i	$-0.403627\pi$
$919$	18.0775	0.596322	0.298161	+	0.954516i	$0.403627\pi$
$920$	0	0
$921$	−8.73972	−0.287983
$922$	0	0
$923$	− 0.116369i	− 0.00383033i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	10.7699i	0.353730i
$928$	0	0
$929$	−8.93003	−0.292985	−0.146492	−	0.989212i	$-0.546798\pi$
$929$	−8.93003	−0.292985	−0.146492	+	0.989212i	$0.546798\pi$
$930$	0	0
$931$	6.72565	0.220424
$932$	0	0
$933$	− 15.4561i	− 0.506011i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 0.0607147i	− 0.00198346i	−1.00000		0.000991732i	$-0.999684\pi$
$937$	− 0.0607147i	− 0.00198346i	1.00000		0.000991732i	$-0.000315678\pi$
$938$	0	0
$939$	−1.55898	−0.0508752
$940$	0	0
$941$	20.6449	0.673005	0.336503	−	0.941682i	$-0.390756\pi$
$941$	20.6449	0.673005	0.336503	+	0.941682i	$0.390756\pi$
$942$	0	0
$943$	75.7249i	2.46594i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	51.3845i	1.66977i	0.550423	+	0.834886i	$0.314466\pi$
$947$	51.3845i	1.66977i	−0.550423	+	0.834886i	$0.685534\pi$
$948$	0	0
$949$	5.04948	0.163913
$950$	0	0
$951$	5.46256	0.177136
$952$	0	0
$953$	− 24.5442i	− 0.795064i	−0.917588	−	0.397532i	$-0.869867\pi$
$953$	− 24.5442i	− 0.795064i	0.917588	−	0.397532i	$-0.130133\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 7.24467i	− 0.234187i
$958$	0	0
$959$	−19.3747	−0.625643
$960$	0	0
$961$	60.4253	1.94920
$962$	0	0
$963$	4.34344i	0.139965i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	38.1844i	1.22793i	0.789334	+	0.613964i	$0.210426\pi$
$967$	38.1844i	1.22793i	−0.789334	+	0.613964i	$0.789574\pi$
$968$	0	0
$969$	−6.33484	−0.203504
$970$	0	0
$971$	25.8172	0.828512	0.414256	−	0.910160i	$-0.364042\pi$
$971$	25.8172	0.828512	0.414256	+	0.910160i	$0.364042\pi$
$972$	0	0
$973$	23.8237i	0.763753i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20.6259i	0.659881i	0.944002	+	0.329941i	$0.107029\pi$
$977$	20.6259i	0.659881i	−0.944002	+	0.329941i	$0.892971\pi$
$978$	0	0
$979$	11.2813	0.360551
$980$	0	0
$981$	1.07127	0.0342030
$982$	0	0
$983$	22.1336i	0.705953i	0.935632	+	0.352976i	$0.114830\pi$
$983$	22.1336i	0.705953i	−0.935632	+	0.352976i	$0.885170\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 14.7278i	− 0.468792i
$988$	0	0
$989$	−24.8445	−0.790008
$990$	0	0
$991$	−1.66509	−0.0528933	−0.0264466	−	0.999650i	$-0.508419\pi$
$991$	−1.66509	−0.0528933	−0.0264466	+	0.999650i	$0.508419\pi$
$992$	0	0
$993$	0.855804i	0.0271581i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 23.4762i	− 0.743497i	−0.928333	−	0.371749i	$-0.878758\pi$
$997$	− 23.4762i	− 0.743497i	0.928333	−	0.371749i	$-0.121242\pi$
$998$	0	0
$999$	5.36700	0.169804

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.d.c.1249.5		8
5.2	odd	4		7500.2.a.f.1.4		4
5.3	odd	4		7500.2.a.e.1.1		4
5.4	even	2	inner	7500.2.d.c.1249.4		8
25.3	odd	20		300.2.m.b.241.1	yes	8
25.4	even	10		1500.2.o.b.49.4		16
25.6	even	5		1500.2.o.b.949.3		16
25.8	odd	20		300.2.m.b.61.1	✓	8
25.17	odd	20		1500.2.m.a.301.2		8
25.19	even	10		1500.2.o.b.949.2		16
25.21	even	5		1500.2.o.b.49.1		16
25.22	odd	20		1500.2.m.a.1201.2		8
75.8	even	20		900.2.n.b.361.2		8
75.53	even	20		900.2.n.b.541.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.m.b.61.1	✓	8	25.8	odd	20
300.2.m.b.241.1	yes	8	25.3	odd	20
900.2.n.b.361.2		8	75.8	even	20
900.2.n.b.541.2		8	75.53	even	20
1500.2.m.a.301.2		8	25.17	odd	20
1500.2.m.a.1201.2		8	25.22	odd	20
1500.2.o.b.49.1		16	25.21	even	5
1500.2.o.b.49.4		16	25.4	even	10
1500.2.o.b.949.2		16	25.19	even	10
1500.2.o.b.949.3		16	25.6	even	5
7500.2.a.e.1.1		4	5.3	odd	4
7500.2.a.f.1.4		4	5.2	odd	4
7500.2.d.c.1249.4		8	5.4	even	2	inner
7500.2.d.c.1249.5		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.74037i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -1.74037i q^{7} -1.00000 q^{9} +2.32027 q^{11} -4.93831i q^{13} +3.74037i q^{17} +1.69364 q^{19} +1.74037 q^{21} -8.67867i q^{23} -1.00000i q^{27} -3.12233 q^{29} +9.56166 q^{31} +2.32027i q^{33} +5.36700i q^{37} +4.93831 q^{39} -8.72540 q^{41} -2.86270i q^{43} -8.46248i q^{47} +3.97112 q^{49} -3.74037 q^{51} +1.34449i q^{53} +1.69364i q^{57} -4.38793 q^{59} -15.3316 q^{61} +1.74037i q^{63} +9.23075i q^{67} +8.67867 q^{69} +0.0235645 q^{71} +1.02251i q^{73} -4.03813i q^{77} +0.798776 q^{79} +1.00000 q^{81} -2.37008i q^{83} -3.12233i q^{87} +4.86205 q^{89} -8.59447 q^{91} +9.56166i q^{93} -3.46151i q^{97} -2.32027 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} - 2 q^{11} + 10 q^{19} - 8 q^{21} - 12 q^{29} + 22 q^{31} + 10 q^{39} + 8 q^{49} - 8 q^{51} - 2 q^{59} - 44 q^{61} + 18 q^{69} + 40 q^{71} + 6 q^{79} + 8 q^{81} + 30 q^{89} - 20 q^{91} + 2 q^{99}+O(q^{100}) \) 8 * q - 8 * q^9 - 2 * q^11 + 10 * q^19 - 8 * q^21 - 12 * q^29 + 22 * q^31 + 10 * q^39 + 8 * q^49 - 8 * q^51 - 2 * q^59 - 44 * q^61 + 18 * q^69 + 40 * q^71 + 6 * q^79 + 8 * q^81 + 30 * q^89 - 20 * q^91 + 2 * q^99

Introduction

L-functions

Modular forms

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Database

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