LMFDB - Embedded newform 304.7.e.f.113.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,7,Mod(113,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("304.113");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	304.e (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:	$69.9364414204$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.19
Character	$\chi$	$=$	304.113
Dual form			304.7.e.f.113.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.4888i q^{3} -77.3334 q^{5} +256.225 q^{7} +597.007 q^{9} +O(q^{10})$	$q+11.4888i q^{3} -77.3334 q^{5} +256.225 q^{7} +597.007 q^{9} -1884.83 q^{11} -823.968i q^{13} -888.468i q^{15} +231.875 q^{17} +(559.006 + 6836.18i) q^{19} +2943.72i q^{21} +19868.2 q^{23} -9644.55 q^{25} +15234.2i q^{27} -11124.2i q^{29} -40963.0i q^{31} -21654.4i q^{33} -19814.8 q^{35} +26804.4i q^{37} +9466.41 q^{39} -31976.1i q^{41} +148300. q^{43} -46168.6 q^{45} +780.447 q^{47} -51997.5 q^{49} +2663.96i q^{51} +229656. i q^{53} +145760. q^{55} +(-78539.5 + 6422.31i) q^{57} +291584. i q^{59} -127065. q^{61} +152968. q^{63} +63720.3i q^{65} +436488. i q^{67} +228262. i q^{69} -186679. i q^{71} -458991. q^{73} -110804. i q^{75} -482941. q^{77} +356689. i q^{79} +260195. q^{81} +564945. q^{83} -17931.7 q^{85} +127803. q^{87} -1.09641e6i q^{89} -211122. i q^{91} +470616. q^{93} +(-43229.8 - 528665. i) q^{95} +1.23494e6i q^{97} -1.12526e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 720 q^{7} - 8670 q^{9}+O(q^{10}) $ 30 * q + 720 * q^7 - 8670 * q^9	$ 30 q + 720 q^{7} - 8670 q^{9} + 2524 q^{11} + 9700 q^{17} - 4014 q^{19} + 39376 q^{23} + 110742 q^{25} + 19976 q^{35} + 266500 q^{39} + 106788 q^{43} - 91360 q^{45} - 222756 q^{47} + 593586 q^{49} - 540936 q^{55} - 545972 q^{57} - 242640 q^{61} + 377716 q^{63} + 545964 q^{73} - 272356 q^{77} + 2189926 q^{81} - 1542652 q^{83} - 826908 q^{85} + 2729572 q^{87} - 2139912 q^{93} - 2142716 q^{95} + 293012 q^{99}+O(q^{100}) $ 30 * q + 720 * q^7 - 8670 * q^9 + 2524 * q^11 + 9700 * q^17 - 4014 * q^19 + 39376 * q^23 + 110742 * q^25 + 19976 * q^35 + 266500 * q^39 + 106788 * q^43 - 91360 * q^45 - 222756 * q^47 + 593586 * q^49 - 540936 * q^55 - 545972 * q^57 - 242640 * q^61 + 377716 * q^63 + 545964 * q^73 - 272356 * q^77 + 2189926 * q^81 - 1542652 * q^83 - 826908 * q^85 + 2729572 * q^87 - 2139912 * q^93 - 2142716 * q^95 + 293012 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$1$	$1$

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
$3$			11.4888i			0.425511i	0.977105	+	0.212756i	$0.0682438\pi$
$3$			11.4888i			0.425511i	−0.977105	+	0.212756i	$0.931756\pi$
$4$		0			0
$5$	−77.3334			−0.618667			−0.309334	−	0.950954i	$-0.600106\pi$
$5$	−77.3334			−0.618667			−0.309334	+	0.950954i	$0.600106\pi$
$6$		0			0
$7$	256.225			0.747013			0.373506	−	0.927628i	$-0.378155\pi$
$7$	256.225			0.747013			0.373506	+	0.927628i	$0.378155\pi$
$8$		0			0
$9$	597.007			0.818940
$10$		0			0
$11$	−1884.83			−1.41610			−0.708050	−	0.706162i	$-0.750425\pi$
$11$	−1884.83			−1.41610			−0.708050	+	0.706162i	$0.750425\pi$
$12$		0			0
$13$		−	823.968i		−	0.375042i	−0.982261	−	0.187521i	$-0.939955\pi$
$13$		−	823.968i		−	0.375042i	0.982261	−	0.187521i	$-0.0600453\pi$
$14$		0			0
$15$		−	888.468i		−	0.263250i
$16$		0			0
$17$	231.875			0.0471962			0.0235981	−	0.999722i	$-0.492488\pi$
$17$	231.875			0.0471962			0.0235981	+	0.999722i	$0.492488\pi$
$18$		0			0
$19$	559.006	+	6836.18i	0.0814996	+	0.996673i
$19$	559.006	+	6836.18i	0.0814996	+	0.996673i
$20$		0			0
$21$			2943.72i			0.317862i
$22$		0			0
$23$	19868.2			1.63296			0.816478	−	0.577376i	$-0.195923\pi$
$23$	19868.2			1.63296			0.816478	+	0.577376i	$0.195923\pi$
$24$		0			0
$25$	−9644.55			−0.617251
$26$		0			0
$27$			15234.2i			0.773979i
$28$		0			0
$29$		−	11124.2i		−	0.456114i	−0.973648	−	0.228057i	$-0.926763\pi$
$29$		−	11124.2i		−	0.456114i	0.973648	−	0.228057i	$-0.0732373\pi$
$30$		0			0
$31$		−	40963.0i		−	1.37501i	−0.726178	−	0.687507i	$-0.758705\pi$
$31$		−	40963.0i		−	1.37501i	0.726178	−	0.687507i	$-0.241295\pi$
$32$		0			0
$33$		−	21654.4i		−	0.602566i
$34$		0			0
$35$	−19814.8			−0.462152
$36$		0			0
$37$			26804.4i			0.529178i	0.964361	+	0.264589i	$0.0852362\pi$
$37$			26804.4i			0.529178i	−0.964361	+	0.264589i	$0.914764\pi$
$38$		0			0
$39$	9466.41			0.159585
$40$		0			0
$41$		−	31976.1i		−	0.463953i	−0.972721	−	0.231977i	$-0.925481\pi$
$41$		−	31976.1i		−	0.463953i	0.972721	−	0.231977i	$-0.0745193\pi$
$42$		0			0
$43$	148300.			1.86524			0.932620	−	0.360861i	$-0.117517\pi$
$43$	148300.			1.86524			0.932620	+	0.360861i	$0.117517\pi$
$44$		0			0
$45$	−46168.6			−0.506651
$46$		0			0
$47$	780.447			0.00751709			0.00375855	−	0.999993i	$-0.498804\pi$
$47$	780.447			0.00751709			0.00375855	+	0.999993i	$0.498804\pi$
$48$		0			0
$49$	−51997.5			−0.441972
$50$		0			0
$51$			2663.96i			0.0200825i
$52$		0			0
$53$			229656.i			1.54259i	0.636480	+	0.771294i	$0.280390\pi$
$53$			229656.i			1.54259i	−0.636480	+	0.771294i	$0.719610\pi$
$54$		0			0
$55$	145760.			0.876095
$56$		0			0
$57$	−78539.5	+	6422.31i	−0.424096	+	0.0346790i
$58$		0			0
$59$			291584.i			1.41973i	0.704336	+	0.709867i	$0.251245\pi$
$59$			291584.i			1.41973i	−0.704336	+	0.709867i	$0.748755\pi$
$60$		0			0
$61$	−127065.			−0.559805			−0.279902	−	0.960028i	$-0.590302\pi$
$61$	−127065.			−0.559805			−0.279902	+	0.960028i	$0.590302\pi$
$62$		0			0
$63$	152968.			0.611759
$64$		0			0
$65$			63720.3i			0.232026i
$66$		0			0
$67$			436488.i			1.45127i	0.688080	+	0.725635i	$0.258454\pi$
$67$			436488.i			1.45127i	−0.688080	+	0.725635i	$0.741546\pi$
$68$		0			0
$69$			228262.i			0.694841i
$70$		0			0
$71$		−	186679.i		−	0.521578i	−0.965396	−	0.260789i	$-0.916017\pi$
$71$		−	186679.i		−	0.521578i	0.965396	−	0.260789i	$-0.0839827\pi$
$72$		0			0
$73$	−458991.			−1.17987			−0.589937	−	0.807450i	$-0.700847\pi$
$73$	−458991.			−1.17987			−0.589937	+	0.807450i	$0.700847\pi$
$74$		0			0
$75$		−	110804.i		−	0.262647i
$76$		0			0
$77$	−482941.			−1.05785
$78$		0			0
$79$			356689.i			0.723449i	0.932285	+	0.361725i	$0.117812\pi$
$79$			356689.i			0.723449i	−0.932285	+	0.361725i	$0.882188\pi$
$80$		0			0
$81$	260195.			0.489604
$82$		0			0
$83$	564945.			0.988034			0.494017	−	0.869452i	$-0.335528\pi$
$83$	564945.			0.988034			0.494017	+	0.869452i	$0.335528\pi$
$84$		0			0
$85$	−17931.7			−0.0291987
$86$		0			0
$87$	127803.			0.194082
$88$		0			0
$89$		−	1.09641e6i		−	1.55527i	−0.628719	−	0.777633i	$-0.716420\pi$
$89$		−	1.09641e6i		−	1.55527i	0.628719	−	0.777633i	$-0.283580\pi$
$90$		0			0
$91$		−	211122.i		−	0.280162i
$92$		0			0
$93$	470616.			0.585084
$94$		0			0
$95$	−43229.8	−	528665.i	−0.0504212	−	0.616609i
$96$		0			0
$97$			1.23494e6i			1.35310i	0.736395	+	0.676552i	$0.236526\pi$
$97$			1.23494e6i			1.35310i	−0.736395	+	0.676552i	$0.763474\pi$
$98$		0			0
$99$	−1.12526e6			−1.15970
$100$		0			0
$101$	−902691.			−0.876143			−0.438071	−	0.898940i	$-0.644338\pi$
$101$	−902691.			−0.876143			−0.438071	+	0.898940i	$0.644338\pi$
$102$		0			0
$103$			1.82609e6i			1.67113i	0.549394	+	0.835563i	$0.314859\pi$
$103$			1.82609e6i			1.67113i	−0.549394	+	0.835563i	$0.685141\pi$
$104$		0			0
$105$		−	227648.i		−	0.196651i
$106$		0			0
$107$			395473.i			0.322823i	0.986887	+	0.161412i	$0.0516047\pi$
$107$			395473.i			0.322823i	−0.986887	+	0.161412i	$0.948395\pi$
$108$		0			0
$109$			2.56541e6i			1.98097i	0.137629	+	0.990484i	$0.456052\pi$
$109$			2.56541e6i			1.98097i	−0.137629	+	0.990484i	$0.543948\pi$
$110$		0			0
$111$	−307951.			−0.225171
$112$		0			0
$113$			1.02258e6i			0.708701i	0.935113	+	0.354351i	$0.115298\pi$
$113$			1.02258e6i			0.708701i	−0.935113	+	0.354351i	$0.884702\pi$
$114$		0			0
$115$	−1.53647e6			−1.01026
$116$		0			0
$117$		−	491915.i		−	0.307137i
$118$		0			0
$119$	59412.2			0.0352561
$120$		0			0
$121$	1.78102e6			1.00534
$122$		0			0
$123$	367367.			0.197417
$124$		0			0
$125$	1.95418e6			1.00054
$126$		0			0
$127$			3.41288e6i			1.66614i	0.553171	+	0.833068i	$0.313418\pi$
$127$			3.41288e6i			1.66614i	−0.553171	+	0.833068i	$0.686582\pi$
$128$		0			0
$129$			1.70378e6i			0.793680i
$130$		0			0
$131$	636368.			0.283070			0.141535	−	0.989933i	$-0.454796\pi$
$131$	636368.			0.283070			0.141535	+	0.989933i	$0.454796\pi$
$132$		0			0
$133$	143232.	+	1.75160e6i	0.0608813	+	0.744528i
$134$		0			0
$135$		−	1.17812e6i		−	0.478836i
$136$		0			0
$137$	−2.21449e6			−0.861215			−0.430608	−	0.902539i	$-0.641701\pi$
$137$	−2.21449e6			−0.861215			−0.430608	+	0.902539i	$0.641701\pi$
$138$		0			0
$139$	−3.27484e6			−1.21940			−0.609700	−	0.792632i	$-0.708710\pi$
$139$	−3.27484e6			−1.21940			−0.609700	+	0.792632i	$0.708710\pi$
$140$		0			0
$141$			8966.40i			0.00319861i
$142$		0			0
$143$			1.55304e6i			0.531098i
$144$		0			0
$145$			860270.i			0.282183i
$146$		0			0
$147$		−	597389.i		−	0.188064i
$148$		0			0
$149$	3.31805e6			1.00305			0.501526	−	0.865142i	$-0.332772\pi$
$149$	3.31805e6			1.00305			0.501526	+	0.865142i	$0.332772\pi$
$150$		0			0
$151$			2.12259e6i			0.616502i	0.951305	+	0.308251i	$0.0997436\pi$
$151$			2.12259e6i			0.616502i	−0.951305	+	0.308251i	$0.900256\pi$
$152$		0			0
$153$	138431.			0.0386508
$154$		0			0
$155$			3.16781e6i			0.850676i
$156$		0			0
$157$	1.02033e6			0.263658			0.131829	−	0.991273i	$-0.457915\pi$
$157$	1.02033e6			0.263658			0.131829	+	0.991273i	$0.457915\pi$
$158$		0			0
$159$	−2.63847e6			−0.656388
$160$		0			0
$161$	5.09073e6			1.21984
$162$		0			0
$163$	−880998.			−0.203429			−0.101714	−	0.994814i	$-0.532433\pi$
$163$	−880998.			−0.203429			−0.101714	+	0.994814i	$0.532433\pi$
$164$		0			0
$165$			1.67461e6i			0.372788i
$166$		0			0
$167$		−	4.94116e6i		−	1.06091i	−0.847712	−	0.530456i	$-0.822021\pi$
$167$		−	4.94116e6i		−	1.06091i	0.847712	−	0.530456i	$-0.177979\pi$
$168$		0			0
$169$	4.14789e6			0.859343
$170$		0			0
$171$	333731.	+	4.08125e6i	0.0667433	+	0.816216i
$172$		0			0
$173$			3.30772e6i			0.638837i	0.947614	+	0.319418i	$0.103488\pi$
$173$			3.30772e6i			0.638837i	−0.947614	+	0.319418i	$0.896512\pi$
$174$		0			0
$175$	−2.47118e6			−0.461094
$176$		0			0
$177$	−3.34994e6			−0.604113
$178$		0			0
$179$			4.07676e6i			0.710814i	0.934712	+	0.355407i	$0.115658\pi$
$179$			4.07676e6i			0.710814i	−0.934712	+	0.355407i	$0.884342\pi$
$180$		0			0
$181$		−	5.76217e6i		−	0.971740i	−0.874031	−	0.485870i	$-0.838503\pi$
$181$		−	5.76217e6i		−	0.971740i	0.874031	−	0.485870i	$-0.161497\pi$
$182$		0			0
$183$		−	1.45982e6i		−	0.238203i
$184$		0			0
$185$		−	2.07288e6i		−	0.327385i
$186$		0			0
$187$	−437044.			−0.0668345
$188$		0			0
$189$			3.90340e6i			0.578172i
$190$		0			0
$191$	1.99933e6			0.286936			0.143468	−	0.989655i	$-0.454175\pi$
$191$	1.99933e6			0.286936			0.143468	+	0.989655i	$0.454175\pi$
$192$		0			0
$193$			5.89342e6i			0.819777i	0.912136	+	0.409889i	$0.134432\pi$
$193$			5.89342e6i			0.819777i	−0.912136	+	0.409889i	$0.865568\pi$
$194$		0			0
$195$	−732069.			−0.0987298
$196$		0			0
$197$	−1.33091e7			−1.74081			−0.870403	−	0.492340i	$-0.836142\pi$
$197$	−1.33091e7			−1.74081			−0.870403	+	0.492340i	$0.836142\pi$
$198$		0			0
$199$	1.24111e7			1.57490			0.787448	−	0.616382i	$-0.211402\pi$
$199$	1.24111e7			1.57490			0.787448	+	0.616382i	$0.211402\pi$
$200$		0			0
$201$	−5.01472e6			−0.617531
$202$		0			0
$203$		−	2.85030e6i		−	0.340723i
$204$		0			0
$205$			2.47282e6i			0.287033i
$206$		0			0
$207$	1.18615e7			1.33729
$208$		0			0
$209$	−1.05363e6	−	1.28850e7i	−0.115412	−	1.41139i
$210$		0			0
$211$		−	1.41553e7i		−	1.50686i	−0.657531	−	0.753428i	$-0.728399\pi$
$211$		−	1.41553e7i		−	1.50686i	0.657531	−	0.753428i	$-0.271601\pi$
$212$		0			0
$213$	2.14471e6			0.221937
$214$		0			0
$215$	−1.14685e7			−1.15396
$216$		0			0
$217$		−	1.04958e7i		−	1.02715i
$218$		0			0
$219$		−	5.27325e6i		−	0.502049i
$220$		0			0
$221$		−	191057.i		−	0.0177006i
$222$		0			0
$223$			1.13723e7i			1.02549i	0.858540	+	0.512747i	$0.171372\pi$
$223$			1.13723e7i			1.02549i	−0.858540	+	0.512747i	$0.828628\pi$
$224$		0			0
$225$	−5.75787e6			−0.505492
$226$		0			0
$227$		−	9.29381e6i		−	0.794541i	−0.917702	−	0.397270i	$-0.869958\pi$
$227$		−	9.29381e6i		−	0.794541i	0.917702	−	0.397270i	$-0.130042\pi$
$228$		0			0
$229$	8.96069e6			0.746165			0.373082	−	0.927798i	$-0.378301\pi$
$229$	8.96069e6			0.746165			0.373082	+	0.927798i	$0.378301\pi$
$230$		0			0
$231$		−	5.54841e6i		−	0.450125i
$232$		0			0
$233$	−5.08687e6			−0.402145			−0.201073	−	0.979576i	$-0.564443\pi$
$233$	−5.08687e6			−0.402145			−0.201073	+	0.979576i	$0.564443\pi$
$234$		0			0
$235$	−60354.6			−0.00465058
$236$		0			0
$237$	−4.09793e6			−0.307836
$238$		0			0
$239$	4.27935e6			0.313462			0.156731	−	0.987641i	$-0.449904\pi$
$239$	4.27935e6			0.313462			0.156731	+	0.987641i	$0.449904\pi$
$240$		0			0
$241$			4.01681e6i			0.286966i	0.989653	+	0.143483i	$0.0458303\pi$
$241$			4.01681e6i			0.286966i	−0.989653	+	0.143483i	$0.954170\pi$
$242$		0			0
$243$			1.40951e7i			0.982311i
$244$		0			0
$245$	4.02115e6			0.273433
$246$		0			0
$247$	5.63280e6	−	460603.i	0.373795	−	0.0305658i
$248$		0			0
$249$			6.49054e6i			0.420420i
$250$		0			0
$251$	5.65888e6			0.357857			0.178928	−	0.983862i	$-0.442737\pi$
$251$	5.65888e6			0.357857			0.178928	+	0.983862i	$0.442737\pi$
$252$		0			0
$253$	−3.74481e7			−2.31243
$254$		0			0
$255$		−	206013.i		−	0.0124244i
$256$		0			0
$257$			9.13037e6i			0.537885i	0.963156	+	0.268942i	$0.0866741\pi$
$257$			9.13037e6i			0.537885i	−0.963156	+	0.268942i	$0.913326\pi$
$258$		0			0
$259$			6.86798e6i			0.395302i
$260$		0			0
$261$		−	6.64121e6i		−	0.373530i
$262$		0			0
$263$	2.86830e6			0.157673			0.0788366	−	0.996888i	$-0.474879\pi$
$263$	2.86830e6			0.157673			0.0788366	+	0.996888i	$0.474879\pi$
$264$		0			0
$265$		−	1.77601e7i		−	0.954348i
$266$		0			0
$267$	1.25965e7			0.661783
$268$		0			0
$269$		−	1.08355e7i		−	0.556660i	−0.960485	−	0.278330i	$-0.910219\pi$
$269$		−	1.08355e7i		−	0.556660i	0.960485	−	0.278330i	$-0.0897809\pi$
$270$		0			0
$271$	−1.47673e7			−0.741980			−0.370990	−	0.928637i	$-0.620982\pi$
$271$	−1.47673e7			−0.741980			−0.370990	+	0.928637i	$0.620982\pi$
$272$		0			0
$273$	2.42553e6			0.119212
$274$		0			0
$275$	1.81783e7			0.874089
$276$		0			0
$277$	1.41485e7			0.665689			0.332845	−	0.942982i	$-0.391992\pi$
$277$	1.41485e7			0.665689			0.332845	+	0.942982i	$0.391992\pi$
$278$		0			0
$279$		−	2.44552e7i		−	1.12605i
$280$		0			0
$281$		−	3.14411e7i		−	1.41703i	−0.705696	−	0.708515i	$-0.749366\pi$
$281$		−	3.14411e7i		−	1.41703i	0.705696	−	0.708515i	$-0.250634\pi$
$282$		0			0
$283$	−726237.			−0.0320420			−0.0160210	−	0.999872i	$-0.505100\pi$
$283$	−726237.			−0.0320420			−0.0160210	+	0.999872i	$0.505100\pi$
$284$		0			0
$285$	6.07373e6	−	496659.i	0.262374	−	0.0214548i
$286$		0			0
$287$		−	8.19310e6i		−	0.346579i
$288$		0			0
$289$	−2.40838e7			−0.997773
$290$		0			0
$291$	−1.41880e7			−0.575761
$292$		0			0
$293$			1.10079e7i			0.437623i	0.975767	+	0.218812i	$0.0702180\pi$
$293$			1.10079e7i			0.437623i	−0.975767	+	0.218812i	$0.929782\pi$
$294$		0			0
$295$		−	2.25491e7i		−	0.878343i
$296$		0			0
$297$		−	2.87139e7i		−	1.09603i
$298$		0			0
$299$		−	1.63707e7i		−	0.612428i
$300$		0			0
$301$	3.79981e7			1.39336
$302$		0			0
$303$		−	1.03708e7i		−	0.372808i
$304$		0			0
$305$	9.82637e6			0.346333
$306$		0			0
$307$		−	2.53316e7i		−	0.875482i	−0.899101	−	0.437741i	$-0.855779\pi$
$307$		−	2.53316e7i		−	0.875482i	0.899101	−	0.437741i	$-0.144221\pi$
$308$		0			0
$309$	−2.09795e7			−0.711083
$310$		0			0
$311$	2.53975e7			0.844324			0.422162	−	0.906520i	$-0.361271\pi$
$311$	2.53975e7			0.844324			0.422162	+	0.906520i	$0.361271\pi$
$312$		0			0
$313$	5.53186e7			1.80401			0.902003	−	0.431730i	$-0.142097\pi$
$313$	5.53186e7			1.80401			0.902003	+	0.431730i	$0.142097\pi$
$314$		0			0
$315$	−1.18296e7			−0.378475
$316$		0			0
$317$		−	4.31314e7i		−	1.35399i	−0.735987	−	0.676996i	$-0.763281\pi$
$317$		−	4.31314e7i		−	1.35399i	0.735987	−	0.676996i	$-0.236719\pi$
$318$		0			0
$319$			2.09672e7i			0.645904i
$320$		0			0
$321$	−4.54350e6			−0.137365
$322$		0			0
$323$	129619.	+	1.58514e6i	0.00384647	+	0.0470392i
$324$		0			0
$325$			7.94680e6i			0.231495i
$326$		0			0
$327$	−2.94735e7			−0.842924
$328$		0			0
$329$	199970.			0.00561536
$330$		0			0
$331$		−	1.69061e7i		−	0.466185i	−0.972455	−	0.233093i	$-0.925115\pi$
$331$		−	1.69061e7i		−	0.466185i	0.972455	−	0.233093i	$-0.0748845\pi$
$332$		0			0
$333$			1.60024e7i			0.433365i
$334$		0			0
$335$		−	3.37551e7i		−	0.897853i
$336$		0			0
$337$		−	3.50007e7i		−	0.914508i	−0.889336	−	0.457254i	$-0.848833\pi$
$337$		−	3.50007e7i		−	0.914508i	0.889336	−	0.457254i	$-0.151167\pi$
$338$		0			0
$339$	−1.17483e7			−0.301560
$340$		0			0
$341$			7.72084e7i			1.94716i
$342$		0			0
$343$	−4.34678e7			−1.07717
$344$		0			0
$345$		−	1.76522e7i		−	0.429875i
$346$		0			0
$347$	−2.95189e7			−0.706500			−0.353250	−	0.935529i	$-0.614924\pi$
$347$	−2.95189e7			−0.706500			−0.353250	+	0.935529i	$0.614924\pi$
$348$		0			0
$349$	5.25810e7			1.23695			0.618475	−	0.785804i	$-0.287751\pi$
$349$	5.25810e7			1.23695			0.618475	+	0.785804i	$0.287751\pi$
$350$		0			0
$351$	1.25525e7			0.290275
$352$		0			0
$353$	6.15626e7			1.39956			0.699782	−	0.714356i	$-0.253280\pi$
$353$	6.15626e7			1.39956			0.699782	+	0.714356i	$0.253280\pi$
$354$		0			0
$355$			1.44365e7i			0.322683i
$356$		0			0
$357$			682575.i			0.0150019i
$358$		0			0
$359$	−4.20339e6			−0.0908483			−0.0454241	−	0.998968i	$-0.514464\pi$
$359$	−4.20339e6			−0.0908483			−0.0454241	+	0.998968i	$0.514464\pi$
$360$		0			0
$361$	−4.64209e7	+	7.64294e6i	−0.986716	+	0.162457i
$362$		0			0
$363$			2.04618e7i			0.427783i
$364$		0			0
$365$	3.54953e7			0.729949
$366$		0			0
$367$	−3.11729e7			−0.630635			−0.315318	−	0.948986i	$-0.602111\pi$
$367$	−3.11729e7			−0.630635			−0.315318	+	0.948986i	$0.602111\pi$
$368$		0			0
$369$		−	1.90900e7i		−	0.379950i
$370$		0			0
$371$			5.88436e7i			1.15233i
$372$		0			0
$373$			9.30221e7i			1.79250i	0.443547	+	0.896251i	$0.353720\pi$
$373$			9.30221e7i			1.79250i	−0.443547	+	0.896251i	$0.646280\pi$
$374$		0			0
$375$			2.24512e7i			0.425741i
$376$		0			0
$377$	−9.16596e6			−0.171062
$378$		0			0
$379$		−	1.15436e7i		−	0.212042i	−0.994364	−	0.106021i	$-0.966189\pi$
$379$		−	1.15436e7i		−	0.212042i	0.994364	−	0.106021i	$-0.0338111\pi$
$380$		0			0
$381$	−3.92099e7			−0.708959
$382$		0			0
$383$			4.36243e7i			0.776483i	0.921558	+	0.388242i	$0.126917\pi$
$383$			4.36243e7i			0.776483i	−0.921558	+	0.388242i	$0.873083\pi$
$384$		0			0
$385$	3.73475e7			0.654454
$386$		0			0
$387$	8.85360e7			1.52752
$388$		0			0
$389$	−8.70917e7			−1.47954			−0.739772	−	0.672857i	$-0.765067\pi$
$389$	−8.70917e7			−1.47954			−0.739772	+	0.672857i	$0.765067\pi$
$390$		0			0
$391$	4.60693e6			0.0770693
$392$		0			0
$393$			7.31110e6i			0.120450i
$394$		0			0
$395$		−	2.75840e7i		−	0.447574i
$396$		0			0
$397$	5.15204e7			0.823395			0.411697	−	0.911321i	$-0.364936\pi$
$397$	5.15204e7			0.823395			0.411697	+	0.911321i	$0.364936\pi$
$398$		0			0
$399$	−2.01238e7	+	1.64556e6i	−0.316805	+	0.0259057i
$400$		0			0
$401$		−	1.11044e8i		−	1.72211i	−0.508509	−	0.861057i	$-0.669803\pi$
$401$		−	1.11044e8i		−	1.72211i	0.508509	−	0.861057i	$-0.330197\pi$
$402$		0			0
$403$	−3.37523e7			−0.515689
$404$		0			0
$405$	−2.01218e7			−0.302902
$406$		0			0
$407$		−	5.05218e7i		−	0.749369i
$408$		0			0
$409$		−	5.16367e7i		−	0.754725i	−0.926066	−	0.377362i	$-0.876831\pi$
$409$		−	5.16367e7i		−	0.754725i	0.926066	−	0.377362i	$-0.123169\pi$
$410$		0			0
$411$		−	2.54418e7i		−	0.366457i
$412$		0			0
$413$			7.47111e7i			1.06056i
$414$		0			0
$415$	−4.36891e7			−0.611264
$416$		0			0
$417$		−	3.76240e7i		−	0.518868i
$418$		0			0
$419$	−3.26913e7			−0.444417			−0.222208	−	0.974999i	$-0.571327\pi$
$419$	−3.26913e7			−0.444417			−0.222208	+	0.974999i	$0.571327\pi$
$420$		0			0
$421$			1.09657e7i			0.146957i	0.997297	+	0.0734783i	$0.0234100\pi$
$421$			1.09657e7i			0.146957i	−0.997297	+	0.0734783i	$0.976590\pi$
$422$		0			0
$423$	465933.			0.00615605
$424$		0			0
$425$	−2.23633e6			−0.0291319
$426$		0			0
$427$	−3.25573e7			−0.418181
$428$		0			0
$429$	−1.78426e7			−0.225988
$430$		0			0
$431$			7.39494e7i			0.923640i	0.886974	+	0.461820i	$0.152803\pi$
$431$			7.39494e7i			0.923640i	−0.886974	+	0.461820i	$0.847197\pi$
$432$		0			0
$433$		−	1.15721e7i		−	0.142544i	−0.997457	−	0.0712718i	$-0.977294\pi$
$433$		−	1.15721e7i		−	0.142544i	0.997457	−	0.0712718i	$-0.0227058\pi$
$434$		0			0
$435$	−9.88347e6			−0.120072
$436$		0			0
$437$	1.11064e7	+	1.35822e8i	0.133085	+	1.62752i
$438$		0			0
$439$			6.78269e7i			0.801694i	0.916145	+	0.400847i	$0.131284\pi$
$439$			6.78269e7i			0.801694i	−0.916145	+	0.400847i	$0.868716\pi$
$440$		0			0
$441$	−3.10429e7			−0.361949
$442$		0			0
$443$	1.24923e8			1.43691			0.718457	−	0.695572i	$-0.244849\pi$
$443$	1.24923e8			1.43691			0.718457	+	0.695572i	$0.244849\pi$
$444$		0			0
$445$			8.47894e7i			0.962192i
$446$		0			0
$447$			3.81204e7i			0.426810i
$448$		0			0
$449$			1.67999e6i			0.0185596i	0.999957	+	0.00927978i	$0.00295389\pi$
$449$			1.67999e6i			0.0185596i	−0.999957	+	0.00927978i	$0.997046\pi$
$450$		0			0
$451$			6.02696e7i			0.657005i
$452$		0			0
$453$	−2.43860e7			−0.262328
$454$		0			0
$455$			1.63268e7i			0.173327i
$456$		0			0
$457$	9.04772e7			0.947962			0.473981	−	0.880535i	$-0.342817\pi$
$457$	9.04772e7			0.947962			0.473981	+	0.880535i	$0.342817\pi$
$458$		0			0
$459$			3.53244e6i			0.0365289i
$460$		0			0
$461$	−1.19579e8			−1.22054			−0.610270	−	0.792193i	$-0.708939\pi$
$461$	−1.19579e8			−1.22054			−0.610270	+	0.792193i	$0.708939\pi$
$462$		0			0
$463$	3.32272e7			0.334773			0.167387	−	0.985891i	$-0.446467\pi$
$463$	3.32272e7			0.334773			0.167387	+	0.985891i	$0.446467\pi$
$464$		0			0
$465$	−3.63944e7			−0.361972
$466$		0			0
$467$	6.55256e7			0.643370			0.321685	−	0.946847i	$-0.395751\pi$
$467$	6.55256e7			0.643370			0.321685	+	0.946847i	$0.395751\pi$
$468$		0			0
$469$			1.11839e8i			1.08412i
$470$		0			0
$471$			1.17223e7i			0.112189i
$472$		0			0
$473$	−2.79519e8			−2.64137
$474$		0			0
$475$	−5.39136e6	−	6.59319e7i	−0.0503057	−	0.615198i
$476$		0			0
$477$			1.37106e8i			1.26329i
$478$		0			0
$479$	1.30819e8			1.19032			0.595161	−	0.803607i	$-0.297088\pi$
$479$	1.30819e8			1.19032			0.595161	+	0.803607i	$0.297088\pi$
$480$		0			0
$481$	2.20860e7			0.198464
$482$		0			0
$483$			5.84864e7i			0.519055i
$484$		0			0
$485$		−	9.55022e7i		−	0.837121i
$486$		0			0
$487$			1.29425e8i			1.12055i	0.828307	+	0.560275i	$0.189305\pi$
$487$			1.29425e8i			1.12055i	−0.828307	+	0.560275i	$0.810695\pi$
$488$		0			0
$489$		−	1.01216e7i		−	0.0865612i
$490$		0			0
$491$	5.29569e7			0.447381			0.223691	−	0.974660i	$-0.428189\pi$
$491$	5.29569e7			0.447381			0.223691	+	0.974660i	$0.428189\pi$
$492$		0			0
$493$		−	2.57941e6i		−	0.0215268i
$494$		0			0
$495$	8.70200e7			0.717469
$496$		0			0
$497$		−	4.78318e7i		−	0.389626i
$498$		0			0
$499$	−7.72723e7			−0.621902			−0.310951	−	0.950426i	$-0.600648\pi$
$499$	−7.72723e7			−0.621902			−0.310951	+	0.950426i	$0.600648\pi$
$500$		0			0
$501$	5.67680e7			0.451430
$502$		0			0
$503$	7.52061e7			0.590948			0.295474	−	0.955351i	$-0.404522\pi$
$503$	7.52061e7			0.590948			0.295474	+	0.955351i	$0.404522\pi$
$504$		0			0
$505$	6.98081e7			0.542041
$506$		0			0
$507$			4.76542e7i			0.365660i
$508$		0			0
$509$			1.29903e8i			0.985068i	0.870293	+	0.492534i	$0.163929\pi$
$509$			1.29903e8i			0.985068i	−0.870293	+	0.492534i	$0.836071\pi$
$510$		0			0
$511$	−1.17605e8			−0.881381
$512$		0			0
$513$	−1.04144e8	+	8.51603e6i	−0.771405	+	0.0630790i
$514$		0			0
$515$		−	1.41217e8i		−	1.03387i
$516$		0			0
$517$	−1.47101e6			−0.0106450
$518$		0			0
$519$	−3.80017e7			−0.271832
$520$		0			0
$521$			4.14232e7i			0.292908i	0.989217	+	0.146454i	$0.0467860\pi$
$521$			4.14232e7i			0.292908i	−0.989217	+	0.146454i	$0.953214\pi$
$522$		0			0
$523$		−	5.03013e7i		−	0.351620i	−0.984424	−	0.175810i	$-0.943745\pi$
$523$		−	5.03013e7i		−	0.351620i	0.984424	−	0.175810i	$-0.0562545\pi$
$524$		0			0
$525$		−	2.83909e7i		−	0.196201i
$526$		0			0
$527$		−	9.49830e6i		−	0.0648954i
$528$		0			0
$529$	2.46709e8			1.66655
$530$		0			0
$531$			1.74078e8i			1.16268i
$532$		0			0
$533$	−2.63473e7			−0.174002
$534$		0			0
$535$		−	3.05832e7i		−	0.199720i
$536$		0			0
$537$	−4.68371e7			−0.302459
$538$		0			0
$539$	9.80065e7			0.625876
$540$		0			0
$541$	−8.01116e7			−0.505945			−0.252973	−	0.967473i	$-0.581408\pi$
$541$	−8.01116e7			−0.505945			−0.252973	+	0.967473i	$0.581408\pi$
$542$		0			0
$543$	6.62004e7			0.413486
$544$		0			0
$545$		−	1.98392e8i		−	1.22556i
$546$		0			0
$547$			1.56604e8i			0.956844i	0.878130	+	0.478422i	$0.158791\pi$
$547$			1.56604e8i			0.956844i	−0.878130	+	0.478422i	$0.841209\pi$
$548$		0			0
$549$	−7.58588e7			−0.458447
$550$		0			0
$551$	7.60469e7	−	6.21848e6i	0.454597	−	0.0371732i
$552$		0			0
$553$			9.13927e7i			0.540426i
$554$		0			0
$555$	2.38149e7			0.139306
$556$		0			0
$557$	4.77912e7			0.276556			0.138278	−	0.990393i	$-0.455843\pi$
$557$	4.77912e7			0.276556			0.138278	+	0.990393i	$0.455843\pi$
$558$		0			0
$559$		−	1.22194e8i		−	0.699544i
$560$		0			0
$561$		−	5.02112e6i		−	0.0284388i
$562$		0			0
$563$			2.09041e8i			1.17140i	0.810528	+	0.585700i	$0.199180\pi$
$563$			2.09041e8i			1.17140i	−0.810528	+	0.585700i	$0.800820\pi$
$564$		0			0
$565$		−	7.90798e7i		−	0.438450i
$566$		0			0
$567$	6.66687e7			0.365740
$568$		0			0
$569$			1.66768e7i			0.0905264i	0.998975	+	0.0452632i	$0.0144127\pi$
$569$			1.66768e7i			0.0905264i	−0.998975	+	0.0452632i	$0.985587\pi$
$570$		0			0
$571$	−1.37693e6			−0.00739612			−0.00369806	−	0.999993i	$-0.501177\pi$
$571$	−1.37693e6			−0.00739612			−0.00369806	+	0.999993i	$0.501177\pi$
$572$		0			0
$573$			2.29699e7i			0.122094i
$574$		0			0
$575$	−1.91620e8			−1.00794
$576$		0			0
$577$	−1.08556e8			−0.565103			−0.282551	−	0.959252i	$-0.591181\pi$
$577$	−1.08556e8			−0.565103			−0.282551	+	0.959252i	$0.591181\pi$
$578$		0			0
$579$	−6.77084e7			−0.348824
$580$		0			0
$581$	1.44753e8			0.738074
$582$		0			0
$583$		−	4.32862e8i		−	2.18446i
$584$		0			0
$585$			3.80415e7i			0.190016i
$586$		0			0
$587$	−3.45910e8			−1.71021			−0.855103	−	0.518458i	$-0.826506\pi$
$587$	−3.45910e8			−1.71021			−0.855103	+	0.518458i	$0.826506\pi$
$588$		0			0
$589$	2.80031e8	−	2.28986e7i	1.37044	−	0.112063i
$590$		0			0
$591$		−	1.52906e8i		−	0.740732i
$592$		0			0
$593$	1.10072e8			0.527851			0.263925	−	0.964543i	$-0.414983\pi$
$593$	1.10072e8			0.527851			0.263925	+	0.964543i	$0.414983\pi$
$594$		0			0
$595$	−4.59455e6			−0.0218118
$596$		0			0
$597$			1.42589e8i			0.670135i
$598$		0			0
$599$			3.05915e8i			1.42338i	0.702494	+	0.711690i	$0.252070\pi$
$599$			3.05915e8i			1.42338i	−0.702494	+	0.711690i	$0.747930\pi$
$600$		0			0
$601$		−	1.14924e8i		−	0.529403i	−0.964330	−	0.264702i	$-0.914727\pi$
$601$		−	1.14924e8i		−	0.529403i	0.964330	−	0.264702i	$-0.0852734\pi$
$602$		0			0
$603$			2.60587e8i			1.18850i
$604$		0			0
$605$	−1.37732e8			−0.621971
$606$		0			0
$607$			1.35384e8i			0.605344i	0.953095	+	0.302672i	$0.0978786\pi$
$607$			1.35384e8i			0.605344i	−0.953095	+	0.302672i	$0.902121\pi$
$608$		0			0
$609$	3.27465e7			0.144982
$610$		0			0
$611$		−	643063.i		−	0.00281923i
$612$		0			0
$613$	3.84066e7			0.166734			0.0833670	−	0.996519i	$-0.473433\pi$
$613$	3.84066e7			0.166734			0.0833670	+	0.996519i	$0.473433\pi$
$614$		0			0
$615$	−2.84098e7			−0.122136
$616$		0			0
$617$	2.72584e8			1.16050			0.580249	−	0.814439i	$-0.302955\pi$
$617$	2.72584e8			1.16050			0.580249	+	0.814439i	$0.302955\pi$
$618$		0			0
$619$	−2.38512e8			−1.00563			−0.502816	−	0.864394i	$-0.667703\pi$
$619$	−2.38512e8			−1.00563			−0.502816	+	0.864394i	$0.667703\pi$
$620$		0			0
$621$			3.02676e8i			1.26387i
$622$		0			0
$623$		−	2.80929e8i		−	1.16180i
$624$		0			0
$625$	−427343.			−0.00175040
$626$		0			0
$627$	1.48034e8	−	1.21050e7i	0.600562	−	0.0491089i
$628$		0			0
$629$			6.21527e6i			0.0249752i
$630$		0			0
$631$	−2.22887e8			−0.887148			−0.443574	−	0.896238i	$-0.646290\pi$
$631$	−2.22887e8			−0.887148			−0.443574	+	0.896238i	$0.646290\pi$
$632$		0			0
$633$	1.62627e8			0.641184
$634$		0			0
$635$		−	2.63930e8i		−	1.03078i
$636$		0			0
$637$			4.28443e7i			0.165758i
$638$		0			0
$639$		−	1.11448e8i		−	0.427141i
$640$		0			0
$641$		−	1.39834e8i		−	0.530930i	−0.964120	−	0.265465i	$-0.914474\pi$
$641$		−	1.39834e8i		−	0.530930i	0.964120	−	0.265465i	$-0.0855255\pi$
$642$		0			0
$643$	−3.82057e7			−0.143713			−0.0718563	−	0.997415i	$-0.522892\pi$
$643$	−3.82057e7			−0.143713			−0.0718563	+	0.997415i	$0.522892\pi$
$644$		0			0
$645$		−	1.31759e8i		−	0.491024i
$646$		0			0
$647$	−7.83222e7			−0.289182			−0.144591	−	0.989491i	$-0.546187\pi$
$647$	−7.83222e7			−0.289182			−0.144591	+	0.989491i	$0.546187\pi$
$648$		0			0
$649$		−	5.49585e8i		−	2.01049i
$650$		0			0
$651$	1.20584e8			0.437065
$652$		0			0
$653$	−1.11071e7			−0.0398897			−0.0199448	−	0.999801i	$-0.506349\pi$
$653$	−1.11071e7			−0.0398897			−0.0199448	+	0.999801i	$0.506349\pi$
$654$		0			0
$655$	−4.92125e7			−0.175126
$656$		0			0
$657$	−2.74021e8			−0.966246
$658$		0			0
$659$		−	4.30536e8i		−	1.50437i	−0.658955	−	0.752183i	$-0.729001\pi$
$659$		−	4.30536e8i		−	1.50437i	0.658955	−	0.752183i	$-0.270999\pi$
$660$		0			0
$661$			3.74353e7i			0.129621i	0.997898	+	0.0648107i	$0.0206444\pi$
$661$			3.74353e7i			0.129621i	−0.997898	+	0.0648107i	$0.979356\pi$
$662$		0			0
$663$	2.19502e6			0.00753179
$664$		0			0
$665$	−1.10766e7	−	1.35457e8i	−0.0376653	−	0.460615i
$666$		0			0
$667$		−	2.21017e8i		−	0.744815i
$668$		0			0
$669$	−1.30654e8			−0.436359
$670$		0			0
$671$	2.39496e8			0.792740
$672$		0			0
$673$			1.99594e7i			0.0654789i	0.999464	+	0.0327395i	$0.0104232\pi$
$673$			1.99594e7i			0.0654789i	−0.999464	+	0.0327395i	$0.989577\pi$
$674$		0			0
$675$		−	1.46927e8i		−	0.477739i
$676$		0			0
$677$		−	1.78337e8i		−	0.574745i	−0.957819	−	0.287373i	$-0.907218\pi$
$677$		−	1.78337e8i		−	0.574745i	0.957819	−	0.287373i	$-0.0927818\pi$
$678$		0			0
$679$			3.16423e8i			1.01079i
$680$		0			0
$681$	1.06775e8			0.338086
$682$		0			0
$683$		−	5.00079e8i		−	1.56956i	−0.619777	−	0.784778i	$-0.712777\pi$
$683$		−	5.00079e8i		−	1.56956i	0.619777	−	0.784778i	$-0.287223\pi$
$684$		0			0
$685$	1.71254e8			0.532806
$686$		0			0
$687$			1.02948e8i			0.317501i
$688$		0			0
$689$	1.89229e8			0.578536
$690$		0			0
$691$	−6.32624e8			−1.91739			−0.958697	−	0.284430i	$-0.908196\pi$
$691$	−6.32624e8			−1.91739			−0.958697	+	0.284430i	$0.908196\pi$
$692$		0			0
$693$	−2.88320e8			−0.866312
$694$		0			0
$695$	2.53255e8			0.754403
$696$		0			0
$697$		−	7.41446e6i		−	0.0218968i
$698$		0			0
$699$		−	5.84420e7i		−	0.171117i
$700$		0			0
$701$	1.46737e8			0.425977			0.212988	−	0.977055i	$-0.431680\pi$
$701$	1.46737e8			0.425977			0.212988	+	0.977055i	$0.431680\pi$
$702$		0			0
$703$	−1.83240e8	+	1.49838e7i	−0.527417	+	0.0431278i
$704$		0			0
$705$		−	693402.i		−	0.00197887i
$706$		0			0
$707$	−2.31292e8			−0.654490
$708$		0			0
$709$	8.20359e7			0.230179			0.115089	−	0.993355i	$-0.463285\pi$
$709$	8.20359e7			0.230179			0.115089	+	0.993355i	$0.463285\pi$
$710$		0			0
$711$			2.12946e8i			0.592462i
$712$		0			0
$713$		−	8.13861e8i		−	2.24534i
$714$		0			0
$715$		−	1.20102e8i		−	0.328573i
$716$		0			0
$717$			4.91646e7i			0.133381i
$718$		0			0
$719$	−1.15430e8			−0.310551			−0.155275	−	0.987871i	$-0.549626\pi$
$719$	−1.15430e8			−0.310551			−0.155275	+	0.987871i	$0.549626\pi$
$720$		0			0
$721$			4.67889e8i			1.24835i
$722$		0			0
$723$	−4.61484e7			−0.122107
$724$		0			0
$725$			1.07288e8i			0.281537i
$726$		0			0
$727$	2.90928e8			0.757150			0.378575	−	0.925571i	$-0.376414\pi$
$727$	2.90928e8			0.757150			0.378575	+	0.925571i	$0.376414\pi$
$728$		0			0
$729$	2.77468e7			0.0716192
$730$		0			0
$731$	3.43869e7			0.0880322
$732$		0			0
$733$	3.92514e8			0.996650			0.498325	−	0.866990i	$-0.333949\pi$
$733$	3.92514e8			0.996650			0.498325	+	0.866990i	$0.333949\pi$
$734$		0			0
$735$			4.61981e7i			0.116349i
$736$		0			0
$737$		−	8.22706e8i		−	2.05514i
$738$		0			0
$739$	6.28357e8			1.55695			0.778473	−	0.627679i	$-0.215995\pi$
$739$	6.28357e8			1.55695			0.778473	+	0.627679i	$0.215995\pi$
$740$		0			0
$741$	5.29178e6	+	6.47141e7i	0.0130061	+	0.159054i
$742$		0			0
$743$		−	4.74636e8i		−	1.15716i	−0.815625	−	0.578581i	$-0.803606\pi$
$743$		−	4.74636e8i		−	1.15716i	0.815625	−	0.578581i	$-0.196394\pi$
$744$		0			0
$745$	−2.56596e8			−0.620556
$746$		0			0
$747$	3.37276e8			0.809141
$748$		0			0
$749$			1.01330e8i			0.241153i
$750$		0			0
$751$			9.02586e7i			0.213093i	0.994308	+	0.106546i	$0.0339793\pi$
$751$			9.02586e7i			0.213093i	−0.994308	+	0.106546i	$0.966021\pi$
$752$		0			0
$753$			6.50137e7i			0.152272i
$754$		0			0
$755$		−	1.64147e8i		−	0.381409i
$756$		0			0
$757$	−4.67280e8			−1.07718			−0.538592	−	0.842567i	$-0.681044\pi$
$757$	−4.67280e8			−1.07718			−0.538592	+	0.842567i	$0.681044\pi$
$758$		0			0
$759$		−	4.30234e8i		−	0.983965i
$760$		0			0
$761$	−7.85218e8			−1.78171			−0.890854	−	0.454290i	$-0.849893\pi$
$761$	−7.85218e8			−1.78171			−0.890854	+	0.454290i	$0.849893\pi$
$762$		0			0
$763$			6.57323e8i			1.47981i
$764$		0			0
$765$	−1.07053e7			−0.0239120
$766$		0			0
$767$	2.40256e8			0.532460
$768$		0			0
$769$	6.45064e8			1.41848			0.709240	−	0.704967i	$-0.249038\pi$
$769$	6.45064e8			1.41848			0.709240	+	0.704967i	$0.249038\pi$
$770$		0			0
$771$	−1.04897e8			−0.228876
$772$		0			0
$773$		−	5.52757e8i		−	1.19673i	−0.801224	−	0.598365i	$-0.795817\pi$
$773$		−	5.52757e8i		−	1.19673i	0.801224	−	0.598365i	$-0.204183\pi$
$774$		0			0
$775$			3.95070e8i			0.848729i
$776$		0			0
$777$	−7.89048e7			−0.168206
$778$		0			0
$779$	2.18595e8	−	1.78749e7i	0.462410	−	0.0378120i
$780$		0			0
$781$			3.51857e8i			0.738607i
$782$		0			0
$783$	1.69468e8			0.353023
$784$		0			0
$785$	−7.89053e7			−0.163116
$786$		0			0
$787$		−	4.80916e8i		−	0.986608i	−0.869857	−	0.493304i	$-0.835789\pi$
$787$		−	4.80916e8i		−	0.986608i	0.869857	−	0.493304i	$-0.164211\pi$
$788$		0			0
$789$			3.29534e7i			0.0670917i
$790$		0			0
$791$			2.62012e8i			0.529409i
$792$		0			0
$793$			1.04698e8i			0.209951i
$794$		0			0
$795$	2.04042e8			0.406086
$796$		0			0
$797$			6.14406e8i			1.21361i	0.794850	+	0.606807i	$0.207550\pi$
$797$			6.14406e8i			1.21361i	−0.794850	+	0.606807i	$0.792450\pi$
$798$		0			0
$799$	180966.			0.000354778
$800$		0			0
$801$		−	6.54567e8i		−	1.27367i
$802$		0			0
$803$	8.65119e8			1.67082
$804$		0			0
$805$	−3.93684e8			−0.754675
$806$		0			0
$807$	1.24486e8			0.236865
$808$		0			0
$809$	−7.51916e8			−1.42012			−0.710058	−	0.704143i	$-0.751331\pi$
$809$	−7.51916e8			−1.42012			−0.710058	+	0.704143i	$0.751331\pi$
$810$		0			0
$811$			3.85314e8i			0.722358i	0.932497	+	0.361179i	$0.117626\pi$
$811$			3.85314e8i			0.722358i	−0.932497	+	0.361179i	$0.882374\pi$
$812$		0			0
$813$		−	1.69658e8i		−	0.315721i
$814$		0			0
$815$	6.81306e7			0.125855
$816$		0			0
$817$	8.29004e7	+	1.01380e9i	0.152016	+	1.85903i
$818$		0			0
$819$		−	1.26041e8i		−	0.229436i
$820$		0			0
$821$	−8.40003e8			−1.51793			−0.758965	−	0.651132i	$-0.774294\pi$
$821$	−8.40003e8			−1.51793			−0.758965	+	0.651132i	$0.774294\pi$
$822$		0			0
$823$	−6.50021e8			−1.16608			−0.583040	−	0.812444i	$-0.698137\pi$
$823$	−6.50021e8			−1.16608			−0.583040	+	0.812444i	$0.698137\pi$
$824$		0			0
$825$			2.08847e8i			0.371935i
$826$		0			0
$827$			1.15568e8i			0.204326i	0.994768	+	0.102163i	$0.0325763\pi$
$827$			1.15568e8i			0.204326i	−0.994768	+	0.102163i	$0.967424\pi$
$828$		0			0
$829$			9.68211e7i			0.169944i	0.996383	+	0.0849721i	$0.0270801\pi$
$829$			9.68211e7i			0.169944i	−0.996383	+	0.0849721i	$0.972920\pi$
$830$		0			0
$831$			1.62549e8i			0.283258i
$832$		0			0
$833$	−1.20569e7			−0.0208594
$834$		0			0
$835$			3.82117e8i			0.656352i
$836$		0			0
$837$	6.24041e8			1.06423
$838$		0			0
$839$		−	1.97743e8i		−	0.334822i	−0.985887	−	0.167411i	$-0.946459\pi$
$839$		−	1.97743e8i		−	0.334822i	0.985887	−	0.167411i	$-0.0535407\pi$
$840$		0			0
$841$	4.71076e8			0.791960
$842$		0			0
$843$	3.61221e8			0.602962
$844$		0			0
$845$	−3.20770e8			−0.531647
$846$		0			0
$847$	4.56343e8			0.751002
$848$		0			0
$849$		−	8.34359e6i		−	0.0136342i
$850$		0			0
$851$			5.32555e8i			0.864124i
$852$		0			0
$853$	3.76486e8			0.606599			0.303299	−	0.952895i	$-0.401912\pi$
$853$	3.76486e8			0.606599			0.303299	+	0.952895i	$0.401912\pi$
$854$		0			0
$855$	−2.58085e7	−	3.15617e8i	−0.0412919	−	0.504966i
$856$		0			0
$857$			1.13903e8i			0.180965i	0.995898	+	0.0904824i	$0.0288409\pi$
$857$			1.13903e8i			0.180965i	−0.995898	+	0.0904824i	$0.971159\pi$
$858$		0			0
$859$	−1.12658e9			−1.77738			−0.888691	−	0.458506i	$-0.848385\pi$
$859$	−1.12658e9			−1.77738			−0.888691	+	0.458506i	$0.848385\pi$
$860$		0			0
$861$	9.41289e7			0.147473
$862$		0			0
$863$		−	7.99053e8i		−	1.24321i	−0.783332	−	0.621603i	$-0.786482\pi$
$863$		−	7.99053e8i		−	1.24321i	0.783332	−	0.621603i	$-0.213518\pi$
$864$		0			0
$865$		−	2.55797e8i		−	0.395227i
$866$		0			0
$867$		−	2.76694e8i		−	0.424563i
$868$		0			0
$869$		−	6.72298e8i		−	1.02448i
$870$		0			0
$871$	3.59652e8			0.544287
$872$		0			0
$873$			7.37269e8i			1.10811i
$874$		0			0
$875$	5.00711e8			0.747416
$876$		0			0
$877$		−	6.88351e8i		−	1.02050i	−0.860027	−	0.510248i	$-0.829554\pi$
$877$		−	6.88351e8i		−	1.02050i	0.860027	−	0.510248i	$-0.170446\pi$
$878$		0			0
$879$	−1.26467e8			−0.186214
$880$		0			0
$881$	5.17126e8			0.756255			0.378128	−	0.925753i	$-0.376568\pi$
$881$	5.17126e8			0.756255			0.378128	+	0.925753i	$0.376568\pi$
$882$		0			0
$883$	5.22926e8			0.759553			0.379777	−	0.925078i	$-0.376001\pi$
$883$	5.22926e8			0.759553			0.379777	+	0.925078i	$0.376001\pi$
$884$		0			0
$885$	2.59063e8			0.373745
$886$		0			0
$887$			3.89838e8i			0.558615i	0.960202	+	0.279308i	$0.0901049\pi$
$887$			3.89838e8i			0.558615i	−0.960202	+	0.279308i	$0.909895\pi$
$888$		0			0
$889$			8.74467e8i			1.24462i
$890$		0			0
$891$	−4.90424e8			−0.693328
$892$		0			0
$893$	436275.	+	5.33528e6i	0.000612640	+	0.00749208i
$894$		0			0
$895$		−	3.15270e8i		−	0.439757i
$896$		0			0
$897$	1.88080e8			0.260595
$898$		0			0
$899$	−4.55680e8			−0.627164
$900$		0			0
$901$			5.32514e7i			0.0728042i
$902$		0			0
$903$			4.36553e8i			0.592889i
$904$		0			0
$905$			4.45608e8i			0.601184i
$906$		0			0
$907$			1.00139e9i			1.34209i	0.741414	+	0.671047i	$0.234155\pi$
$907$			1.00139e9i			1.34209i	−0.741414	+	0.671047i	$0.765845\pi$
$908$		0			0
$909$	−5.38913e8			−0.717508
$910$		0			0
$911$		−	6.80410e8i		−	0.899944i	−0.893043	−	0.449972i	$-0.851434\pi$
$911$		−	6.80410e8i		−	0.899944i	0.893043	−	0.449972i	$-0.148566\pi$
$912$		0			0
$913$	−1.06483e9			−1.39916
$914$		0			0
$915$			1.12893e8i			0.147368i
$916$		0			0
$917$	1.63054e8			0.211457
$918$		0			0
$919$	−1.25738e9			−1.62002			−0.810008	−	0.586419i	$-0.800537\pi$
$919$	−1.25738e9			−1.62002			−0.810008	+	0.586419i	$0.800537\pi$
$920$		0			0
$921$	2.91029e8			0.372527
$922$		0			0
$923$	−1.53817e8			−0.195614
$924$		0			0
$925$		−	2.58517e8i		−	0.326635i
$926$		0			0
$927$			1.09019e9i			1.36855i
$928$		0			0
$929$	3.96937e8			0.495079			0.247539	−	0.968878i	$-0.420378\pi$
$929$	3.96937e8			0.495079			0.247539	+	0.968878i	$0.420378\pi$
$930$		0			0
$931$	−2.90669e7	−	3.55465e8i	−0.0360205	−	0.440502i
$932$		0			0
$933$			2.91786e8i			0.359269i
$934$		0			0
$935$	3.37981e7			0.0413483
$936$		0			0
$937$	−7.90442e7			−0.0960841			−0.0480420	−	0.998845i	$-0.515298\pi$
$937$	−7.90442e7			−0.0960841			−0.0480420	+	0.998845i	$0.515298\pi$
$938$		0			0
$939$			6.35544e8i			0.767624i
$940$		0			0
$941$		−	1.42039e9i		−	1.70466i	−0.523005	−	0.852329i	$-0.675189\pi$
$941$		−	1.42039e9i		−	1.70466i	0.523005	−	0.852329i	$-0.324811\pi$
$942$		0			0
$943$		−	6.35308e8i		−	0.757616i
$944$		0			0
$945$		−	3.01863e8i		−	0.357696i
$946$		0			0
$947$	−3.45527e8			−0.406848			−0.203424	−	0.979091i	$-0.565207\pi$
$947$	−3.45527e8			−0.406848			−0.203424	+	0.979091i	$0.565207\pi$
$948$		0			0
$949$			3.78194e8i			0.442503i
$950$		0			0
$951$	4.95528e8			0.576139
$952$		0			0
$953$			8.18045e8i			0.945145i	0.881292	+	0.472572i	$0.156675\pi$
$953$			8.18045e8i			0.945145i	−0.881292	+	0.472572i	$0.843325\pi$
$954$		0			0
$955$	−1.54615e8			−0.177518
$956$		0			0
$957$	−2.40888e8			−0.274839
$958$		0			0
$959$	−5.67408e8			−0.643339
$960$		0			0
$961$	−7.90468e8			−0.890664
$962$		0			0
$963$			2.36100e8i			0.264373i
$964$		0			0
$965$		−	4.55759e8i		−	0.507169i
$966$		0			0
$967$	4.60352e8			0.509109			0.254555	−	0.967058i	$-0.418071\pi$
$967$	4.60352e8			0.509109			0.254555	+	0.967058i	$0.418071\pi$
$968$		0			0
$969$	−1.82113e7	+	1.48917e6i	−0.0200157	+	0.00163672i
$970$		0			0
$971$		−	1.62908e9i		−	1.77944i	−0.456506	−	0.889721i	$-0.650899\pi$
$971$		−	1.62908e9i		−	1.77944i	0.456506	−	0.889721i	$-0.349101\pi$
$972$		0			0
$973$	−8.39098e8			−0.910907
$974$		0			0
$975$	−9.12992e7			−0.0985038
$976$		0			0
$977$			1.50126e9i			1.60980i	0.593408	+	0.804902i	$0.297782\pi$
$977$			1.50126e9i			1.60980i	−0.593408	+	0.804902i	$0.702218\pi$
$978$		0			0
$979$			2.06655e9i			2.20241i
$980$		0			0
$981$			1.53157e9i			1.62229i
$982$		0			0
$983$			1.77758e9i			1.87141i	0.352785	+	0.935704i	$0.385235\pi$
$983$			1.77758e9i			1.87141i	−0.352785	+	0.935704i	$0.614765\pi$
$984$		0			0
$985$	1.02924e9			1.07698
$986$		0			0
$987$			2.29742e6i			0.00238940i
$988$		0			0
$989$	2.94644e9			3.04585
$990$		0			0
$991$			2.59530e8i			0.266666i	0.991071	+	0.133333i	$0.0425680\pi$
$991$			2.59530e8i			0.266666i	−0.991071	+	0.133333i	$0.957432\pi$
$992$		0			0
$993$	1.94230e8			0.198367
$994$		0			0
$995$	−9.59794e8			−0.974336
$996$		0			0
$997$	8.22070e8			0.829513			0.414757	−	0.909932i	$-0.363867\pi$
$997$	8.22070e8			0.829513			0.414757	+	0.909932i	$0.363867\pi$
$998$		0			0
$999$	−4.08345e8			−0.409573

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.7.e.f.113.19		30
4.3	odd	2		152.7.e.a.113.12	✓	30
19.18	odd	2	inner	304.7.e.f.113.12		30
76.75	even	2		152.7.e.a.113.19	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.7.e.a.113.12	✓	30	4.3	odd	2
152.7.e.a.113.19	yes	30	76.75	even	2
304.7.e.f.113.12		30	19.18	odd	2	inner
304.7.e.f.113.19		30	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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+11.4888i q^{3} -77.3334 q^{5} +256.225 q^{7} +597.007 q^{9} +O(q^{10})\)	\(q+11.4888i q^{3} -77.3334 q^{5} +256.225 q^{7} +597.007 q^{9} -1884.83 q^{11} -823.968i q^{13} -888.468i q^{15} +231.875 q^{17} +(559.006 + 6836.18i) q^{19} +2943.72i q^{21} +19868.2 q^{23} -9644.55 q^{25} +15234.2i q^{27} -11124.2i q^{29} -40963.0i q^{31} -21654.4i q^{33} -19814.8 q^{35} +26804.4i q^{37} +9466.41 q^{39} -31976.1i q^{41} +148300. q^{43} -46168.6 q^{45} +780.447 q^{47} -51997.5 q^{49} +2663.96i q^{51} +229656. i q^{53} +145760. q^{55} +(-78539.5 + 6422.31i) q^{57} +291584. i q^{59} -127065. q^{61} +152968. q^{63} +63720.3i q^{65} +436488. i q^{67} +228262. i q^{69} -186679. i q^{71} -458991. q^{73} -110804. i q^{75} -482941. q^{77} +356689. i q^{79} +260195. q^{81} +564945. q^{83} -17931.7 q^{85} +127803. q^{87} -1.09641e6i q^{89} -211122. i q^{91} +470616. q^{93} +(-43229.8 - 528665. i) q^{95} +1.23494e6i q^{97} -1.12526e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 720 q^{7} - 8670 q^{9}+O(q^{10}) \) 30 * q + 720 * q^7 - 8670 * q^9	\( 30 q + 720 q^{7} - 8670 q^{9} + 2524 q^{11} + 9700 q^{17} - 4014 q^{19} + 39376 q^{23} + 110742 q^{25} + 19976 q^{35} + 266500 q^{39} + 106788 q^{43} - 91360 q^{45} - 222756 q^{47} + 593586 q^{49} - 540936 q^{55} - 545972 q^{57} - 242640 q^{61} + 377716 q^{63} + 545964 q^{73} - 272356 q^{77} + 2189926 q^{81} - 1542652 q^{83} - 826908 q^{85} + 2729572 q^{87} - 2139912 q^{93} - 2142716 q^{95} + 293012 q^{99}+O(q^{100}) \) 30 * q + 720 * q^7 - 8670 * q^9 + 2524 * q^11 + 9700 * q^17 - 4014 * q^19 + 39376 * q^23 + 110742 * q^25 + 19976 * q^35 + 266500 * q^39 + 106788 * q^43 - 91360 * q^45 - 222756 * q^47 + 593586 * q^49 - 540936 * q^55 - 545972 * q^57 - 242640 * q^61 + 377716 * q^63 + 545964 * q^73 - 272356 * q^77 + 2189926 * q^81 - 1542652 * q^83 - 826908 * q^85 + 2729572 * q^87 - 2139912 * q^93 - 2142716 * q^95 + 293012 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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