LMFDB - Embedded newform 252.12.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,12,Mod(1,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("252.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	252.a (trivial)

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:	yes
Analytic conductor:	$193.622481501$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{37321}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 9330 $ x^2 - x - 9330
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 84)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$97.0932$ of defining polynomial
Character	$\chi$	$=$	252.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-332.797 q^{5} +16807.0 q^{7} +O(q^{10})$	$q-332.797 q^{5} +16807.0 q^{7} +624125. q^{11} +488269. q^{13} +2.43955e6 q^{17} -1.29682e7 q^{19} -5.53263e7 q^{23} -4.87174e7 q^{25} +1.18692e8 q^{29} -1.06418e8 q^{31} -5.59331e6 q^{35} +2.79285e8 q^{37} +3.10060e8 q^{41} -6.02928e8 q^{43} -4.65689e8 q^{47} +2.82475e8 q^{49} +7.58136e8 q^{53} -2.07707e8 q^{55} -3.58137e8 q^{59} +6.68270e9 q^{61} -1.62494e8 q^{65} -1.67987e10 q^{67} +1.86196e10 q^{71} -3.05425e10 q^{73} +1.04897e10 q^{77} +5.46443e9 q^{79} +1.06864e10 q^{83} -8.11875e8 q^{85} +9.11585e10 q^{89} +8.20635e9 q^{91} +4.31576e9 q^{95} -1.57472e11 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 5130 q^{5} + 33614 q^{7}+O(q^{10}) $ 2 * q + 5130 * q^5 + 33614 * q^7	$ 2 q + 5130 q^{5} + 33614 q^{7} + 591030 q^{11} - 713456 q^{13} - 2396682 q^{17} + 3231712 q^{19} - 14460822 q^{23} - 67703350 q^{25} + 114772572 q^{29} - 244258232 q^{31} + 86219910 q^{35} - 406917752 q^{37} - 566385966 q^{41} - 458503304 q^{43} + 31797036 q^{47} + 564950498 q^{49} - 948409776 q^{53} - 388498680 q^{55} + 5075089020 q^{59} - 3562962956 q^{61} - 6727276260 q^{65} - 22171908812 q^{67} + 40168648278 q^{71} - 31432036592 q^{73} + 9933441210 q^{77} - 17990522780 q^{79} + 53996735592 q^{83} - 27231241860 q^{85} + 88152601698 q^{89} - 11991054992 q^{91} + 92812449240 q^{95} - 100130067512 q^{97}+O(q^{100}) $ 2 * q + 5130 * q^5 + 33614 * q^7 + 591030 * q^11 - 713456 * q^13 - 2396682 * q^17 + 3231712 * q^19 - 14460822 * q^23 - 67703350 * q^25 + 114772572 * q^29 - 244258232 * q^31 + 86219910 * q^35 - 406917752 * q^37 - 566385966 * q^41 - 458503304 * q^43 + 31797036 * q^47 + 564950498 * q^49 - 948409776 * q^53 - 388498680 * q^55 + 5075089020 * q^59 - 3562962956 * q^61 - 6727276260 * q^65 - 22171908812 * q^67 + 40168648278 * q^71 - 31432036592 * q^73 + 9933441210 * q^77 - 17990522780 * q^79 + 53996735592 * q^83 - 27231241860 * q^85 + 88152601698 * q^89 - 11991054992 * q^91 + 92812449240 * q^95 - 100130067512 * q^97

Display 100 coefficients 10 coefficients

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−332.797	−0.0476260	−0.0238130	−	0.999716i	$-0.507581\pi$
$5$	−332.797	−0.0476260	−0.0238130	+	0.999716i	$0.507581\pi$
$6$	0	0
$7$	16807.0	0.377964
$7$	16807.0	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	624125.	1.16845	0.584227	−	0.811590i	$-0.301398\pi$
$11$	624125.	1.16845	0.584227	+	0.811590i	$0.301398\pi$
$12$	0	0
$13$	488269.	0.364730	0.182365	−	0.983231i	$-0.441625\pi$
$13$	488269.	0.364730	0.182365	+	0.983231i	$0.441625\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.43955e6	0.416717	0.208358	−	0.978053i	$-0.433188\pi$
$17$	2.43955e6	0.416717	0.208358	+	0.978053i	$0.433188\pi$
$18$	0	0
$19$	−1.29682e7	−1.20153	−0.600764	−	0.799426i	$-0.705137\pi$
$19$	−1.29682e7	−1.20153	−0.600764	+	0.799426i	$0.705137\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.53263e7	−1.79237	−0.896187	−	0.443677i	$-0.853674\pi$
$23$	−5.53263e7	−1.79237	−0.896187	+	0.443677i	$0.853674\pi$
$24$	0	0
$25$	−4.87174e7	−0.997732
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.18692e8	1.07457	0.537283	−	0.843402i	$-0.319451\pi$
$29$	1.18692e8	1.07457	0.537283	+	0.843402i	$0.319451\pi$
$30$	0	0
$31$	−1.06418e8	−0.667617	−0.333809	−	0.942641i	$-0.608334\pi$
$31$	−1.06418e8	−0.667617	−0.333809	+	0.942641i	$0.608334\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−5.59331e6	−0.0180009
$36$	0	0
$37$	2.79285e8	0.662122	0.331061	−	0.943609i	$-0.392593\pi$
$37$	2.79285e8	0.662122	0.331061	+	0.943609i	$0.392593\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.10060e8	0.417959	0.208980	−	0.977920i	$-0.432986\pi$
$41$	3.10060e8	0.417959	0.208980	+	0.977920i	$0.432986\pi$
$42$	0	0
$43$	−6.02928e8	−0.625445	−0.312723	−	0.949845i	$-0.601241\pi$
$43$	−6.02928e8	−0.625445	−0.312723	+	0.949845i	$0.601241\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.65689e8	−0.296181	−0.148091	−	0.988974i	$-0.547313\pi$
$47$	−4.65689e8	−0.296181	−0.148091	+	0.988974i	$0.547313\pi$
$48$	0	0
$49$	2.82475e8	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.58136e8	0.249018	0.124509	−	0.992219i	$-0.460264\pi$
$53$	7.58136e8	0.249018	0.124509	+	0.992219i	$0.460264\pi$
$54$	0	0
$55$	−2.07707e8	−0.0556488
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.58137e8	−0.0652173	−0.0326086	−	0.999468i	$-0.510381\pi$
$59$	−3.58137e8	−0.0652173	−0.0326086	+	0.999468i	$0.510381\pi$
$60$	0	0
$61$	6.68270e9	1.01307	0.506533	−	0.862220i	$-0.330927\pi$
$61$	6.68270e9	1.01307	0.506533	+	0.862220i	$0.330927\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.62494e8	−0.0173706
$66$	0	0
$67$	−1.67987e10	−1.52008	−0.760038	−	0.649879i	$-0.774820\pi$
$67$	−1.67987e10	−1.52008	−0.760038	+	0.649879i	$0.774820\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.86196e10	1.22476	0.612378	−	0.790565i	$-0.290213\pi$
$71$	1.86196e10	1.22476	0.612378	+	0.790565i	$0.290213\pi$
$72$	0	0
$73$	−3.05425e10	−1.72436	−0.862182	−	0.506599i	$-0.830902\pi$
$73$	−3.05425e10	−1.72436	−0.862182	+	0.506599i	$0.830902\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.04897e10	0.441634
$78$	0	0
$79$	5.46443e9	0.199800	0.0999001	−	0.994997i	$-0.468148\pi$
$79$	5.46443e9	0.199800	0.0999001	+	0.994997i	$0.468148\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.06864e10	0.297784	0.148892	−	0.988853i	$-0.452429\pi$
$83$	1.06864e10	0.297784	0.148892	+	0.988853i	$0.452429\pi$
$84$	0	0
$85$	−8.11875e8	−0.0198465
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.11585e10	1.73042	0.865211	−	0.501408i	$-0.167184\pi$
$89$	9.11585e10	1.73042	0.865211	+	0.501408i	$0.167184\pi$
$90$	0	0
$91$	8.20635e9	0.137855
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.31576e9	0.0572240
$96$	0	0
$97$	−1.57472e11	−1.86191	−0.930957	−	0.365129i	$-0.881025\pi$
$97$	−1.57472e11	−1.86191	−0.930957	+	0.365129i	$0.881025\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.38659e10	−0.604647	−0.302323	−	0.953205i	$-0.597762\pi$
$101$	−6.38659e10	−0.604647	−0.302323	+	0.953205i	$0.597762\pi$
$102$	0	0
$103$	−2.62366e10	−0.222999	−0.111499	−	0.993764i	$-0.535565\pi$
$103$	−2.62366e10	−0.222999	−0.111499	+	0.993764i	$0.535565\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.59507e11	1.78870	0.894352	−	0.447364i	$-0.147637\pi$
$107$	2.59507e11	1.78870	0.894352	+	0.447364i	$0.147637\pi$
$108$	0	0
$109$	−4.26302e10	−0.265382	−0.132691	−	0.991157i	$-0.542362\pi$
$109$	−4.26302e10	−0.265382	−0.132691	+	0.991157i	$0.542362\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.75861e11	−1.40851	−0.704253	−	0.709949i	$-0.748718\pi$
$113$	−2.75861e11	−1.40851	−0.704253	+	0.709949i	$0.748718\pi$
$114$	0	0
$115$	1.84124e10	0.0853635
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.10016e10	0.157504
$120$	0	0
$121$	1.04221e11	0.365287
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.24628e10	0.0951439
$126$	0	0
$127$	−5.22424e11	−1.40315	−0.701573	−	0.712597i	$-0.747519\pi$
$127$	−5.22424e11	−1.40315	−0.701573	+	0.712597i	$0.747519\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.30049e11	−0.294520	−0.147260	−	0.989098i	$-0.547045\pi$
$131$	−1.30049e11	−0.294520	−0.147260	+	0.989098i	$0.547045\pi$
$132$	0	0
$133$	−2.17956e11	−0.454135
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.40719e11	−0.249109	−0.124554	−	0.992213i	$-0.539750\pi$
$137$	−1.40719e11	−0.249109	−0.124554	+	0.992213i	$0.539750\pi$
$138$	0	0
$139$	5.70001e11	0.931738	0.465869	−	0.884854i	$-0.345742\pi$
$139$	5.70001e11	0.931738	0.465869	+	0.884854i	$0.345742\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.04741e11	0.426170
$144$	0	0
$145$	−3.95003e10	−0.0511772
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.94199e10	−0.0551286	−0.0275643	−	0.999620i	$-0.508775\pi$
$149$	−4.94199e10	−0.0551286	−0.0275643	+	0.999620i	$0.508775\pi$
$150$	0	0
$151$	3.69064e11	0.382585	0.191293	−	0.981533i	$-0.438732\pi$
$151$	3.69064e11	0.382585	0.191293	+	0.981533i	$0.438732\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.54157e10	0.0317959
$156$	0	0
$157$	2.13831e12	1.78905	0.894525	−	0.447017i	$-0.147514\pi$
$157$	2.13831e12	1.78905	0.894525	+	0.447017i	$0.147514\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−9.29869e11	−0.677454
$162$	0	0
$163$	−1.30447e11	−0.0887980	−0.0443990	−	0.999014i	$-0.514137\pi$
$163$	−1.30447e11	−0.0887980	−0.0443990	+	0.999014i	$0.514137\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.94590e12	−1.15926	−0.579630	−	0.814880i	$-0.696803\pi$
$167$	−1.94590e12	−1.15926	−0.579630	+	0.814880i	$0.696803\pi$
$168$	0	0
$169$	−1.55375e12	−0.866972
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.84672e12	−0.906040	−0.453020	−	0.891500i	$-0.649653\pi$
$173$	−1.84672e12	−0.906040	−0.453020	+	0.891500i	$0.649653\pi$
$174$	0	0
$175$	−8.18793e11	−0.377107
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.35139e12	−1.76985	−0.884924	−	0.465736i	$-0.845790\pi$
$179$	−4.35139e12	−1.76985	−0.884924	+	0.465736i	$0.845790\pi$
$180$	0	0
$181$	2.38193e12	0.911373	0.455686	−	0.890140i	$-0.349394\pi$
$181$	2.38193e12	0.911373	0.455686	+	0.890140i	$0.349394\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−9.29451e10	−0.0315342
$186$	0	0
$187$	1.52259e12	0.486915
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2.61356e12	−0.743960	−0.371980	−	0.928241i	$-0.621321\pi$
$191$	−2.61356e12	−0.743960	−0.371980	+	0.928241i	$0.621321\pi$
$192$	0	0
$193$	−3.20618e12	−0.861832	−0.430916	−	0.902392i	$-0.641809\pi$
$193$	−3.20618e12	−0.861832	−0.430916	+	0.902392i	$0.641809\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.55890e11	0.0374330	0.0187165	−	0.999825i	$-0.494042\pi$
$197$	1.55890e11	0.0374330	0.0187165	+	0.999825i	$0.494042\pi$
$198$	0	0
$199$	5.01290e11	0.113867	0.0569334	−	0.998378i	$-0.481868\pi$
$199$	5.01290e11	0.113867	0.0569334	+	0.998378i	$0.481868\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.99486e12	0.406147
$204$	0	0
$205$	−1.03187e11	−0.0199057
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−8.09376e12	−1.40393
$210$	0	0
$211$	4.40262e12	0.724700	0.362350	−	0.932042i	$-0.381975\pi$
$211$	4.40262e12	0.724700	0.362350	+	0.932042i	$0.381975\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.00652e11	0.0297874
$216$	0	0
$217$	−1.78857e12	−0.252336
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.19116e12	0.151989
$222$	0	0
$223$	−1.11197e13	−1.35025	−0.675126	−	0.737702i	$-0.735911\pi$
$223$	−1.11197e13	−1.35025	−0.675126	+	0.737702i	$0.735911\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−8.72831e12	−0.961143	−0.480571	−	0.876956i	$-0.659571\pi$
$227$	−8.72831e12	−0.961143	−0.480571	+	0.876956i	$0.659571\pi$
$228$	0	0
$229$	−1.44002e13	−1.51103	−0.755516	−	0.655130i	$-0.772614\pi$
$229$	−1.44002e13	−1.51103	−0.755516	+	0.655130i	$0.772614\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.04016e13	0.992294	0.496147	−	0.868238i	$-0.334748\pi$
$233$	1.04016e13	0.992294	0.496147	+	0.868238i	$0.334748\pi$
$234$	0	0
$235$	1.54980e11	0.0141059
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4.89114e11	0.0405716	0.0202858	−	0.999794i	$-0.493542\pi$
$239$	4.89114e11	0.0405716	0.0202858	+	0.999794i	$0.493542\pi$
$240$	0	0
$241$	2.90605e12	0.230255	0.115127	−	0.993351i	$-0.463272\pi$
$241$	2.90605e12	0.230255	0.115127	+	0.993351i	$0.463272\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−9.40068e10	−0.00680371
$246$	0	0
$247$	−6.33196e12	−0.438233
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4.16227e12	−0.263709	−0.131855	−	0.991269i	$-0.542093\pi$
$251$	−4.16227e12	−0.263709	−0.131855	+	0.991269i	$0.542093\pi$
$252$	0	0
$253$	−3.45305e13	−2.09431
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.90221e12	0.161472	0.0807358	−	0.996736i	$-0.474273\pi$
$257$	2.90221e12	0.161472	0.0807358	+	0.996736i	$0.474273\pi$
$258$	0	0
$259$	4.69394e12	0.250259
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.56958e13	−1.25923	−0.629616	−	0.776906i	$-0.716788\pi$
$263$	−2.56958e13	−1.25923	−0.629616	+	0.776906i	$0.716788\pi$
$264$	0	0
$265$	−2.52305e11	−0.0118597
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.12610e12	0.308471	0.154235	−	0.988034i	$-0.450709\pi$
$269$	7.12610e12	0.308471	0.154235	+	0.988034i	$0.450709\pi$
$270$	0	0
$271$	1.21194e13	0.503674	0.251837	−	0.967770i	$-0.418965\pi$
$271$	1.21194e13	0.503674	0.251837	+	0.967770i	$0.418965\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.04057e13	−1.16580
$276$	0	0
$277$	−1.07780e13	−0.397098	−0.198549	−	0.980091i	$-0.563623\pi$
$277$	−1.07780e13	−0.397098	−0.198549	+	0.980091i	$0.563623\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.77132e13	−0.943632	−0.471816	−	0.881697i	$-0.656401\pi$
$281$	−2.77132e13	−0.943632	−0.471816	+	0.881697i	$0.656401\pi$
$282$	0	0
$283$	−4.44376e13	−1.45521	−0.727604	−	0.685997i	$-0.759366\pi$
$283$	−4.44376e13	−1.45521	−0.727604	+	0.685997i	$0.759366\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.21117e12	0.157974
$288$	0	0
$289$	−2.83205e13	−0.826347
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.50346e13	−0.406742	−0.203371	−	0.979102i	$-0.565190\pi$
$293$	−1.50346e13	−0.406742	−0.203371	+	0.979102i	$0.565190\pi$
$294$	0	0
$295$	1.19187e11	0.00310604
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.70141e13	−0.653732
$300$	0	0
$301$	−1.01334e13	−0.236396
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.22398e12	−0.0482483
$306$	0	0
$307$	5.12781e13	1.07318	0.536588	−	0.843844i	$-0.319713\pi$
$307$	5.12781e13	1.07318	0.536588	+	0.843844i	$0.319713\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.77153e13	−0.929984	−0.464992	−	0.885315i	$-0.653943\pi$
$311$	−4.77153e13	−0.929984	−0.464992	+	0.885315i	$0.653943\pi$
$312$	0	0
$313$	−9.07049e12	−0.170662	−0.0853310	−	0.996353i	$-0.527195\pi$
$313$	−9.07049e12	−0.170662	−0.0853310	+	0.996353i	$0.527195\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.58378e13	−0.277887	−0.138944	−	0.990300i	$-0.544371\pi$
$317$	−1.58378e13	−0.277887	−0.138944	+	0.990300i	$0.544371\pi$
$318$	0	0
$319$	7.40787e13	1.25558
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.16365e13	−0.500697
$324$	0	0
$325$	−2.37872e13	−0.363902
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7.82683e12	−0.111946
$330$	0	0
$331$	−4.02308e13	−0.556551	−0.278275	−	0.960501i	$-0.589763\pi$
$331$	−4.02308e13	−0.556551	−0.278275	+	0.960501i	$0.589763\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.59056e12	0.0723951
$336$	0	0
$337$	1.67274e13	0.209635	0.104818	−	0.994491i	$-0.466574\pi$
$337$	1.67274e13	0.209635	0.104818	+	0.994491i	$0.466574\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−6.64184e13	−0.780080
$342$	0	0
$343$	4.74756e12	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.11615e14	−1.19099	−0.595497	−	0.803358i	$-0.703045\pi$
$347$	−1.11615e14	−1.19099	−0.595497	+	0.803358i	$0.703045\pi$
$348$	0	0
$349$	−7.63339e13	−0.789183	−0.394591	−	0.918857i	$-0.629114\pi$
$349$	−7.63339e13	−0.789183	−0.394591	+	0.918857i	$0.629114\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.82139e13	−0.662387	−0.331193	−	0.943563i	$-0.607451\pi$
$353$	−6.82139e13	−0.662387	−0.331193	+	0.943563i	$0.607451\pi$
$354$	0	0
$355$	−6.19654e12	−0.0583302
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.00012e14	−1.77026	−0.885131	−	0.465342i	$-0.845931\pi$
$359$	−2.00012e14	−1.77026	−0.885131	+	0.465342i	$0.845931\pi$
$360$	0	0
$361$	5.16833e13	0.443671
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.01644e13	0.0821245
$366$	0	0
$367$	−7.17743e13	−0.562738	−0.281369	−	0.959600i	$-0.590788\pi$
$367$	−7.17743e13	−0.562738	−0.281369	+	0.959600i	$0.590788\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.27420e13	0.0941198
$372$	0	0
$373$	1.63630e14	1.17345	0.586725	−	0.809786i	$-0.300417\pi$
$373$	1.63630e14	1.17345	0.586725	+	0.809786i	$0.300417\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.79537e13	0.391926
$378$	0	0
$379$	2.66107e14	1.74800	0.874000	−	0.485926i	$-0.161518\pi$
$379$	2.66107e14	1.74800	0.874000	+	0.485926i	$0.161518\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.13688e14	−0.704889	−0.352444	−	0.935833i	$-0.614649\pi$
$383$	−1.13688e14	−0.704889	−0.352444	+	0.935833i	$0.614649\pi$
$384$	0	0
$385$	−3.49093e12	−0.0210333
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.02002e14	1.14983	0.574914	−	0.818214i	$-0.305036\pi$
$389$	2.02002e14	1.14983	0.574914	+	0.818214i	$0.305036\pi$
$390$	0	0
$391$	−1.34971e14	−0.746912
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.81854e12	−0.00951568
$396$	0	0
$397$	5.07645e13	0.258352	0.129176	−	0.991622i	$-0.458767\pi$
$397$	5.07645e13	0.258352	0.129176	+	0.991622i	$0.458767\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.40872e14	0.678471	0.339235	−	0.940702i	$-0.389832\pi$
$401$	1.40872e14	0.678471	0.339235	+	0.940702i	$0.389832\pi$
$402$	0	0
$403$	−5.19609e13	−0.243500
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.74309e14	0.773660
$408$	0	0
$409$	−1.59348e14	−0.688442	−0.344221	−	0.938889i	$-0.611857\pi$
$409$	−1.59348e14	−0.688442	−0.344221	+	0.938889i	$0.611857\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−6.01920e12	−0.0246498
$414$	0	0
$415$	−3.55639e12	−0.0141822
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.42921e14	0.540654	0.270327	−	0.962769i	$-0.412868\pi$
$419$	1.42921e14	0.540654	0.270327	+	0.962769i	$0.412868\pi$
$420$	0	0
$421$	−4.52303e14	−1.66678	−0.833390	−	0.552685i	$-0.813603\pi$
$421$	−4.52303e14	−1.66678	−0.833390	+	0.552685i	$0.813603\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.18849e14	−0.415772
$426$	0	0
$427$	1.12316e14	0.382903
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.10697e14	−1.33014	−0.665069	−	0.746782i	$-0.731598\pi$
$431$	−4.10697e14	−1.33014	−0.665069	+	0.746782i	$0.731598\pi$
$432$	0	0
$433$	−5.85422e14	−1.84835	−0.924177	−	0.381964i	$-0.875248\pi$
$433$	−5.85422e14	−1.84835	−0.924177	+	0.381964i	$0.875248\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.17481e14	2.15359
$438$	0	0
$439$	2.35021e14	0.687941	0.343970	−	0.938981i	$-0.388228\pi$
$439$	2.35021e14	0.687941	0.343970	+	0.938981i	$0.388228\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.97425e14	0.828240	0.414120	−	0.910222i	$-0.364089\pi$
$443$	2.97425e14	0.828240	0.414120	+	0.910222i	$0.364089\pi$
$444$	0	0
$445$	−3.03372e13	−0.0824130
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.58185e14	0.926303	0.463151	−	0.886279i	$-0.346719\pi$
$449$	3.58185e14	0.926303	0.463151	+	0.886279i	$0.346719\pi$
$450$	0	0
$451$	1.93516e14	0.488366
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.73104e12	−0.00656547
$456$	0	0
$457$	3.24682e14	0.761938	0.380969	−	0.924588i	$-0.375590\pi$
$457$	3.24682e14	0.761938	0.380969	+	0.924588i	$0.375590\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.25823e14	1.17621	0.588104	−	0.808785i	$-0.299874\pi$
$461$	5.25823e14	1.17621	0.588104	+	0.808785i	$0.299874\pi$
$462$	0	0
$463$	−2.14443e14	−0.468399	−0.234200	−	0.972188i	$-0.575247\pi$
$463$	−2.14443e14	−0.468399	−0.234200	+	0.972188i	$0.575247\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.17232e14	0.869231	0.434615	−	0.900616i	$-0.356884\pi$
$467$	4.17232e14	0.869231	0.434615	+	0.900616i	$0.356884\pi$
$468$	0	0
$469$	−2.82336e14	−0.574534
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.76303e14	−0.730804
$474$	0	0
$475$	6.31775e14	1.19880
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4.41816e14	0.800564	0.400282	−	0.916392i	$-0.368912\pi$
$479$	4.41816e14	0.800564	0.400282	+	0.916392i	$0.368912\pi$
$480$	0	0
$481$	1.36366e14	0.241496
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.24062e13	0.0886754
$486$	0	0
$487$	2.33529e14	0.386306	0.193153	−	0.981169i	$-0.438129\pi$
$487$	2.33529e14	0.386306	0.193153	+	0.981169i	$0.438129\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7.14039e14	−1.12921	−0.564604	−	0.825362i	$-0.690971\pi$
$491$	−7.14039e14	−1.12921	−0.564604	+	0.825362i	$0.690971\pi$
$492$	0	0
$493$	2.89556e14	0.447789
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.12940e14	0.462914
$498$	0	0
$499$	2.17363e14	0.314509	0.157254	−	0.987558i	$-0.449736\pi$
$499$	2.17363e14	0.314509	0.157254	+	0.987558i	$0.449736\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.04538e15	1.44761	0.723806	−	0.690003i	$-0.242391\pi$
$503$	1.04538e15	1.44761	0.723806	+	0.690003i	$0.242391\pi$
$504$	0	0
$505$	2.12544e13	0.0287969
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7.44902e14	0.966388	0.483194	−	0.875513i	$-0.339477\pi$
$509$	7.44902e14	0.966388	0.483194	+	0.875513i	$0.339477\pi$
$510$	0	0
$511$	−5.13328e14	−0.651748
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.73145e12	0.0106205
$516$	0	0
$517$	−2.90648e14	−0.346074
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.72624e14	−0.197012	−0.0985060	−	0.995136i	$-0.531406\pi$
$521$	−1.72624e14	−0.197012	−0.0985060	+	0.995136i	$0.531406\pi$
$522$	0	0
$523$	1.16038e15	1.29670	0.648351	−	0.761341i	$-0.275459\pi$
$523$	1.16038e15	1.29670	0.648351	+	0.761341i	$0.275459\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.59613e14	−0.278207
$528$	0	0
$529$	2.10819e15	2.21260
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.51393e14	0.152442
$534$	0	0
$535$	−8.63631e13	−0.0851888
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.76300e14	0.166922
$540$	0	0
$541$	2.10268e13	0.0195069	0.00975345	−	0.999952i	$-0.496895\pi$
$541$	2.10268e13	0.0195069	0.00975345	+	0.999952i	$0.496895\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.41872e13	0.0126391
$546$	0	0
$547$	−1.15634e15	−1.00961	−0.504805	−	0.863233i	$-0.668436\pi$
$547$	−1.15634e15	−1.00961	−0.504805	+	0.863233i	$0.668436\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.53922e15	−1.29112
$552$	0	0
$553$	9.18407e13	0.0755174
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7.97404e14	−0.630195	−0.315097	−	0.949059i	$-0.602037\pi$
$557$	−7.97404e14	−0.630195	−0.315097	+	0.949059i	$0.602037\pi$
$558$	0	0
$559$	−2.94391e14	−0.228118
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.46261e15	1.08976	0.544881	−	0.838513i	$-0.316575\pi$
$563$	1.46261e15	1.08976	0.544881	+	0.838513i	$0.316575\pi$
$564$	0	0
$565$	9.18056e13	0.0670815
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3.23647e14	−0.227486	−0.113743	−	0.993510i	$-0.536284\pi$
$569$	−3.23647e14	−0.227486	−0.113743	+	0.993510i	$0.536284\pi$
$570$	0	0
$571$	1.95960e15	1.35104	0.675522	−	0.737340i	$-0.263918\pi$
$571$	1.95960e15	1.35104	0.675522	+	0.737340i	$0.263918\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.69535e15	1.78831
$576$	0	0
$577$	−2.85329e14	−0.185729	−0.0928643	−	0.995679i	$-0.529602\pi$
$577$	−2.85329e14	−0.185729	−0.0928643	+	0.995679i	$0.529602\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.79606e14	0.112552
$582$	0	0
$583$	4.73172e14	0.290966
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.17709e15	−0.697109	−0.348554	−	0.937289i	$-0.613327\pi$
$587$	−1.17709e15	−0.697109	−0.348554	+	0.937289i	$0.613327\pi$
$588$	0	0
$589$	1.38005e15	0.802161
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3.97652e14	−0.222691	−0.111346	−	0.993782i	$-0.535516\pi$
$593$	−3.97652e14	−0.222691	−0.111346	+	0.993782i	$0.535516\pi$
$594$	0	0
$595$	−1.36452e13	−0.00750129
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2.73334e15	−1.44826	−0.724130	−	0.689663i	$-0.757758\pi$
$599$	−2.73334e15	−1.44826	−0.724130	+	0.689663i	$0.757758\pi$
$600$	0	0
$601$	4.00554e14	0.208378	0.104189	−	0.994558i	$-0.466775\pi$
$601$	4.00554e14	0.208378	0.104189	+	0.994558i	$0.466775\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.46842e13	−0.0173971
$606$	0	0
$607$	9.35899e14	0.460990	0.230495	−	0.973074i	$-0.425965\pi$
$607$	9.35899e14	0.460990	0.230495	+	0.973074i	$0.425965\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.27382e14	−0.108026
$612$	0	0
$613$	2.71177e15	1.26538	0.632688	−	0.774407i	$-0.281952\pi$
$613$	2.71177e15	1.26538	0.632688	+	0.774407i	$0.281952\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.36175e15	1.51355	0.756775	−	0.653676i	$-0.226774\pi$
$617$	3.36175e15	1.51355	0.756775	+	0.653676i	$0.226774\pi$
$618$	0	0
$619$	−3.33681e15	−1.47582	−0.737909	−	0.674900i	$-0.764187\pi$
$619$	−3.33681e15	−1.47582	−0.737909	+	0.674900i	$0.764187\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.53210e15	0.654038
$624$	0	0
$625$	2.36797e15	0.993200
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.81331e14	0.275918
$630$	0	0
$631$	−8.96218e14	−0.356658	−0.178329	−	0.983971i	$-0.557069\pi$
$631$	−8.96218e14	−0.356658	−0.178329	+	0.983971i	$0.557069\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.73861e14	0.0668262
$636$	0	0
$637$	1.37924e14	0.0521042
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−4.57036e14	−0.166813	−0.0834067	−	0.996516i	$-0.526580\pi$
$641$	−4.57036e14	−0.166813	−0.0834067	+	0.996516i	$0.526580\pi$
$642$	0	0
$643$	−5.40018e15	−1.93753	−0.968764	−	0.247985i	$-0.920232\pi$
$643$	−5.40018e15	−1.93753	−0.968764	+	0.247985i	$0.920232\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.26121e15	0.784092	0.392046	−	0.919946i	$-0.371767\pi$
$647$	2.26121e15	0.784092	0.392046	+	0.919946i	$0.371767\pi$
$648$	0	0
$649$	−2.23522e14	−0.0762035
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.08142e14	0.200439	0.100219	−	0.994965i	$-0.468046\pi$
$653$	6.08142e14	0.200439	0.100219	+	0.994965i	$0.468046\pi$
$654$	0	0
$655$	4.32799e13	0.0140268
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.21738e12	−0.000381556 0	−0.000190778	−	1.00000i	$-0.500061\pi$
$659$	−1.21738e12	−0.000381556 0	−0.000190778		1.00000i	$0.500061\pi$
$660$	0	0
$661$	−2.81659e15	−0.868192	−0.434096	−	0.900867i	$-0.642932\pi$
$661$	−2.81659e15	−0.868192	−0.434096	+	0.900867i	$0.642932\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.25350e13	0.0216286
$666$	0	0
$667$	−6.56679e15	−1.92602
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.17084e15	1.18372
$672$	0	0
$673$	−8.95554e14	−0.250040	−0.125020	−	0.992154i	$-0.539899\pi$
$673$	−8.95554e14	−0.250040	−0.125020	+	0.992154i	$0.539899\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7.02071e15	−1.89733	−0.948667	−	0.316278i	$-0.897567\pi$
$677$	−7.02071e15	−1.89733	−0.948667	+	0.316278i	$0.897567\pi$
$678$	0	0
$679$	−2.64664e15	−0.703737
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−7.32081e15	−1.88471	−0.942357	−	0.334610i	$-0.891396\pi$
$683$	−7.32081e15	−1.88471	−0.942357	+	0.334610i	$0.891396\pi$
$684$	0	0
$685$	4.68308e13	0.0118641
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.70175e14	0.0908241
$690$	0	0
$691$	1.40092e15	0.338285	0.169142	−	0.985592i	$-0.445900\pi$
$691$	1.40092e15	0.338285	0.169142	+	0.985592i	$0.445900\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.89694e14	−0.0443749
$696$	0	0
$697$	7.56407e14	0.174171
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.32171e14	0.0518035	0.0259017	−	0.999664i	$-0.491754\pi$
$701$	2.32171e14	0.0518035	0.0259017	+	0.999664i	$0.491754\pi$
$702$	0	0
$703$	−3.62182e15	−0.795559
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.07339e15	−0.228535
$708$	0	0
$709$	−9.51468e14	−0.199453	−0.0997263	−	0.995015i	$-0.531797\pi$
$709$	−9.51468e14	−0.199453	−0.0997263	+	0.995015i	$0.531797\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.88774e15	1.19662
$714$	0	0
$715$	−1.01417e14	−0.0202968
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8.59883e15	1.66890	0.834450	−	0.551084i	$-0.185786\pi$
$719$	8.59883e15	1.66890	0.834450	+	0.551084i	$0.185786\pi$
$720$	0	0
$721$	−4.40958e14	−0.0842856
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5.78237e15	−1.07213
$726$	0	0
$727$	1.49658e15	0.273313	0.136656	−	0.990619i	$-0.456364\pi$
$727$	1.49658e15	0.273313	0.136656	+	0.990619i	$0.456364\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.47088e15	−0.260634
$732$	0	0
$733$	−2.39572e15	−0.418181	−0.209090	−	0.977896i	$-0.567050\pi$
$733$	−2.39572e15	−0.418181	−0.209090	+	0.977896i	$0.567050\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.04845e16	−1.77614
$738$	0	0
$739$	7.32177e15	1.22200	0.611001	−	0.791630i	$-0.290767\pi$
$739$	7.32177e15	1.22200	0.611001	+	0.791630i	$0.290767\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.00734e16	−1.63207	−0.816034	−	0.578003i	$-0.803832\pi$
$743$	−1.00734e16	−1.63207	−0.816034	+	0.578003i	$0.803832\pi$
$744$	0	0
$745$	1.64468e13	0.00262555
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.36154e15	0.676067
$750$	0	0
$751$	8.54080e15	1.30460	0.652302	−	0.757959i	$-0.273803\pi$
$751$	8.54080e15	1.30460	0.652302	+	0.757959i	$0.273803\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.22823e14	−0.0182210
$756$	0	0
$757$	−9.22341e15	−1.34854	−0.674271	−	0.738484i	$-0.735542\pi$
$757$	−9.22341e15	−1.34854	−0.674271	+	0.738484i	$0.735542\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−8.83238e15	−1.25448	−0.627238	−	0.778828i	$-0.715815\pi$
$761$	−8.83238e15	−1.25448	−0.627238	+	0.778828i	$0.715815\pi$
$762$	0	0
$763$	−7.16486e14	−0.100305
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.74867e14	−0.0237867
$768$	0	0
$769$	−3.03783e15	−0.407350	−0.203675	−	0.979039i	$-0.565289\pi$
$769$	−3.03783e15	−0.407350	−0.203675	+	0.979039i	$0.565289\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−3.21642e15	−0.419165	−0.209583	−	0.977791i	$-0.567211\pi$
$773$	−3.21642e15	−0.419165	−0.209583	+	0.977791i	$0.567211\pi$
$774$	0	0
$775$	5.18443e15	0.666103
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.02091e15	−0.502190
$780$	0	0
$781$	1.16210e16	1.43107
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.11622e14	−0.0852053
$786$	0	0
$787$	−1.26675e16	−1.49566	−0.747828	−	0.663893i	$-0.768903\pi$
$787$	−1.26675e16	−1.49566	−0.747828	+	0.663893i	$0.768903\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−4.63639e15	−0.532365
$792$	0	0
$793$	3.26296e15	0.369495
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.89007e15	0.208189	0.104094	−	0.994567i	$-0.466806\pi$
$797$	1.89007e15	0.208189	0.104094	+	0.994567i	$0.466806\pi$
$798$	0	0
$799$	−1.13607e15	−0.123424
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.90623e16	−2.01484
$804$	0	0
$805$	3.09457e14	0.0322644
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.00839e16	−1.02309	−0.511543	−	0.859257i	$-0.670926\pi$
$809$	−1.00839e16	−1.02309	−0.511543	+	0.859257i	$0.670926\pi$
$810$	0	0
$811$	−1.40098e16	−1.40222	−0.701111	−	0.713052i	$-0.747312\pi$
$811$	−1.40098e16	−1.40222	−0.701111	+	0.713052i	$0.747312\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.34124e13	0.00422909
$816$	0	0
$817$	7.81888e15	0.751490
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.39026e15	0.504339	0.252169	−	0.967683i	$-0.418856\pi$
$821$	5.39026e15	0.504339	0.252169	+	0.967683i	$0.418856\pi$
$822$	0	0
$823$	−1.53505e16	−1.41718	−0.708590	−	0.705621i	$-0.750668\pi$
$823$	−1.53505e16	−1.41718	−0.708590	+	0.705621i	$0.750668\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.95849e15	0.805294	0.402647	−	0.915355i	$-0.368090\pi$
$827$	8.95849e15	0.805294	0.402647	+	0.915355i	$0.368090\pi$
$828$	0	0
$829$	2.03523e16	1.80536	0.902681	−	0.430310i	$-0.141596\pi$
$829$	2.03523e16	1.80536	0.902681	+	0.430310i	$0.141596\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.89113e14	0.0595310
$834$	0	0
$835$	6.47590e14	0.0552109
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.17725e16	−0.977637	−0.488819	−	0.872385i	$-0.662572\pi$
$839$	−1.17725e16	−0.977637	−0.488819	+	0.872385i	$0.662572\pi$
$840$	0	0
$841$	1.88730e15	0.154690
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.17084e14	0.0412904
$846$	0	0
$847$	1.75163e15	0.138065
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.54518e16	−1.18677
$852$	0	0
$853$	−1.64189e16	−1.24487	−0.622436	−	0.782670i	$-0.713857\pi$
$853$	−1.64189e16	−1.24487	−0.622436	+	0.782670i	$0.713857\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.83245e16	1.35406	0.677030	−	0.735955i	$-0.263267\pi$
$857$	1.83245e16	1.35406	0.677030	+	0.735955i	$0.263267\pi$
$858$	0	0
$859$	9.36924e15	0.683505	0.341753	−	0.939790i	$-0.388980\pi$
$859$	9.36924e15	0.683505	0.341753	+	0.939790i	$0.388980\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.68972e16	−1.20159	−0.600793	−	0.799405i	$-0.705148\pi$
$863$	−1.68972e16	−1.20159	−0.600793	+	0.799405i	$0.705148\pi$
$864$	0	0
$865$	6.14582e14	0.0431511
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.41049e15	0.233457
$870$	0	0
$871$	−8.20231e15	−0.554417
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.45602e14	0.0359610
$876$	0	0
$877$	2.14536e16	1.39637	0.698187	−	0.715916i	$-0.253991\pi$
$877$	2.14536e16	1.39637	0.698187	+	0.715916i	$0.253991\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	8.45532e15	0.536738	0.268369	−	0.963316i	$-0.413515\pi$
$881$	8.45532e15	0.536738	0.268369	+	0.963316i	$0.413515\pi$
$882$	0	0
$883$	1.38542e16	0.868557	0.434279	−	0.900779i	$-0.357003\pi$
$883$	1.38542e16	0.868557	0.434279	+	0.900779i	$0.357003\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.33565e16	1.42833	0.714163	−	0.699980i	$-0.246808\pi$
$887$	2.33565e16	1.42833	0.714163	+	0.699980i	$0.246808\pi$
$888$	0	0
$889$	−8.78039e15	−0.530339
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.03913e15	0.355870
$894$	0	0
$895$	1.44813e15	0.0842907
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.26310e16	−0.717398
$900$	0	0
$901$	1.84951e15	0.103770
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.92697e14	−0.0434050
$906$	0	0
$907$	3.95738e15	0.214076	0.107038	−	0.994255i	$-0.465863\pi$
$907$	3.95738e15	0.214076	0.107038	+	0.994255i	$0.465863\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.08505e16	−0.572925	−0.286463	−	0.958091i	$-0.592479\pi$
$911$	−1.08505e16	−0.572925	−0.286463	+	0.958091i	$0.592479\pi$
$912$	0	0
$913$	6.66963e15	0.347947
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.18573e15	−0.111318
$918$	0	0
$919$	−3.36227e16	−1.69199	−0.845994	−	0.533193i	$-0.820992\pi$
$919$	−3.36227e16	−1.69199	−0.845994	+	0.533193i	$0.820992\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9.09139e15	0.446705
$924$	0	0
$925$	−1.36060e16	−0.660620
$926$	0	0
$927$	0	0
$928$	0	0
$929$	2.02417e16	0.959753	0.479877	−	0.877336i	$-0.340681\pi$
$929$	2.02417e16	0.959753	0.479877	+	0.877336i	$0.340681\pi$
$930$	0	0
$931$	−3.66319e15	−0.171647
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5.06711e14	−0.0231898
$936$	0	0
$937$	−1.18132e16	−0.534319	−0.267160	−	0.963652i	$-0.586085\pi$
$937$	−1.18132e16	−0.534319	−0.267160	+	0.963652i	$0.586085\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.11263e16	0.933428	0.466714	−	0.884408i	$-0.345438\pi$
$941$	2.11263e16	0.933428	0.466714	+	0.884408i	$0.345438\pi$
$942$	0	0
$943$	−1.71544e16	−0.749139
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.39757e16	1.02293	0.511466	−	0.859303i	$-0.329103\pi$
$947$	2.39757e16	1.02293	0.511466	+	0.859303i	$0.329103\pi$
$948$	0	0
$949$	−1.49130e16	−0.628927
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.95187e16	1.21643	0.608213	−	0.793774i	$-0.291886\pi$
$953$	2.95187e16	1.21643	0.608213	+	0.793774i	$0.291886\pi$
$954$	0	0
$955$	8.69785e14	0.0354318
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.36506e15	−0.0941543
$960$	0	0
$961$	−1.40836e16	−0.554287
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.06701e15	0.0410456
$966$	0	0
$967$	5.20174e15	0.197835	0.0989174	−	0.995096i	$-0.468462\pi$
$967$	5.20174e15	0.197835	0.0989174	+	0.995096i	$0.468462\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−2.30312e15	−0.0856269	−0.0428134	−	0.999083i	$-0.513632\pi$
$971$	−2.30312e15	−0.0856269	−0.0428134	+	0.999083i	$0.513632\pi$
$972$	0	0
$973$	9.58000e15	0.352164
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.90663e16	1.76345	0.881724	−	0.471765i	$-0.156383\pi$
$977$	4.90663e16	1.76345	0.881724	+	0.471765i	$0.156383\pi$
$978$	0	0
$979$	5.68943e16	2.02192
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.49995e16	0.521233	0.260616	−	0.965442i	$-0.416074\pi$
$983$	1.49995e16	0.521233	0.260616	+	0.965442i	$0.416074\pi$
$984$	0	0
$985$	−5.18798e13	−0.00178278
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.33578e16	1.12103
$990$	0	0
$991$	9.02295e15	0.299877	0.149939	−	0.988695i	$-0.452092\pi$
$991$	9.02295e15	0.299877	0.149939	+	0.988695i	$0.452092\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.66828e14	−0.00542302
$996$	0	0
$997$	6.27970e15	0.201890	0.100945	−	0.994892i	$-0.467813\pi$
$997$	6.27970e15	0.201890	0.100945	+	0.994892i	$0.467813\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.12.a.c.1.1		2
3.2	odd	2		84.12.a.d.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.12.a.d.1.2	✓	2	3.2	odd	2
252.12.a.c.1.1		2	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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	252.a (trivial)

\(f(q)\)	\(=\)	\(q-332.797 q^{5} +16807.0 q^{7} +O(q^{10})\)	\(q-332.797 q^{5} +16807.0 q^{7} +624125. q^{11} +488269. q^{13} +2.43955e6 q^{17} -1.29682e7 q^{19} -5.53263e7 q^{23} -4.87174e7 q^{25} +1.18692e8 q^{29} -1.06418e8 q^{31} -5.59331e6 q^{35} +2.79285e8 q^{37} +3.10060e8 q^{41} -6.02928e8 q^{43} -4.65689e8 q^{47} +2.82475e8 q^{49} +7.58136e8 q^{53} -2.07707e8 q^{55} -3.58137e8 q^{59} +6.68270e9 q^{61} -1.62494e8 q^{65} -1.67987e10 q^{67} +1.86196e10 q^{71} -3.05425e10 q^{73} +1.04897e10 q^{77} +5.46443e9 q^{79} +1.06864e10 q^{83} -8.11875e8 q^{85} +9.11585e10 q^{89} +8.20635e9 q^{91} +4.31576e9 q^{95} -1.57472e11 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5130 q^{5} + 33614 q^{7}+O(q^{10}) \) 2 * q + 5130 * q^5 + 33614 * q^7	\( 2 q + 5130 q^{5} + 33614 q^{7} + 591030 q^{11} - 713456 q^{13} - 2396682 q^{17} + 3231712 q^{19} - 14460822 q^{23} - 67703350 q^{25} + 114772572 q^{29} - 244258232 q^{31} + 86219910 q^{35} - 406917752 q^{37} - 566385966 q^{41} - 458503304 q^{43} + 31797036 q^{47} + 564950498 q^{49} - 948409776 q^{53} - 388498680 q^{55} + 5075089020 q^{59} - 3562962956 q^{61} - 6727276260 q^{65} - 22171908812 q^{67} + 40168648278 q^{71} - 31432036592 q^{73} + 9933441210 q^{77} - 17990522780 q^{79} + 53996735592 q^{83} - 27231241860 q^{85} + 88152601698 q^{89} - 11991054992 q^{91} + 92812449240 q^{95} - 100130067512 q^{97}+O(q^{100}) \) 2 * q + 5130 * q^5 + 33614 * q^7 + 591030 * q^11 - 713456 * q^13 - 2396682 * q^17 + 3231712 * q^19 - 14460822 * q^23 - 67703350 * q^25 + 114772572 * q^29 - 244258232 * q^31 + 86219910 * q^35 - 406917752 * q^37 - 566385966 * q^41 - 458503304 * q^43 + 31797036 * q^47 + 564950498 * q^49 - 948409776 * q^53 - 388498680 * q^55 + 5075089020 * q^59 - 3562962956 * q^61 - 6727276260 * q^65 - 22171908812 * q^67 + 40168648278 * q^71 - 31432036592 * q^73 + 9933441210 * q^77 - 17990522780 * q^79 + 53996735592 * q^83 - 27231241860 * q^85 + 88152601698 * q^89 - 11991054992 * q^91 + 92812449240 * q^95 - 100130067512 * q^97

Introduction

L-functions

Modular forms

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Database

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